A Group Theoretic Approach to Cyclic Cubic Fields

Abstract

Let ${\left(k_{\mu }\right)}_{\mu =1}^4$ be a quartet of cyclic cubic number fields sharing a common conductor $c=pqr$ divisible by exactly three prime(power)s, $p,q,r$. For those components of the quartet whose 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is elementary bicyclic, the automorphism group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$ of the maximal metabelian unramified 3-extension of $k_{\mu }$ is determined by conditions for cubic residue symbols between $p,q,r$ and for ambiguous principal ideals in subfields of the common absolute 3-genus field $k^{*}$ of all $k_{\mu }$. With the aid of the relation rank $d_2\left(\mathfrak{M}\right)$, it is decided whether $\mathfrak{M}$ coincides with the Galois group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })$ of the maximal unramified pro-3-extension of $k_{\mu }$.

1. Introduction

Let k be a cyclic cubic number field, that is, an abelian extension of the rational number field $\mathbb{Q}$ with degree $[k:\mathbb{Q}]=3$ and some positive integer conductor $c>1$ (see Section 2.1). In 1973, Georges Gras [1] determined the rank $\varrho =\varrho \left(k\right)$ of the 3-class group ${\mathrm{Cl}}_3\left(k\right)$ in dependence on the number t of prime(power) divisors $q_1,\dots ,q_t$ of c and on the cubic residue symbols ${\left(\frac{q_i}{q_j}\right)}_3$ for $i\ne j$. For mutual cubic residues, ${\left(\frac{q_i}{q_j}\right)}_3={\left(\frac{q_j}{q_i}\right)}_3=1$, we write $q_i\leftrightarrow q_j$, otherwise $q_i\nleftrightarrow q_j$.

It turned out that $\varrho =0$ for $t=1$, and $\varrho =1$ if $t=2$ and $q_1\nleftrightarrow q_2$. So, in the former case, the maximal unramified pro-3-extension ${\mathrm{F}}_3^{\infty }\left(k\right)$ of k is the base field k itself, and in the latter case, it is the Hilbert 3-class field ${\mathrm{F}}_3^1\left(k\right)$ of k, in fact, $[{\mathrm{F}}_3^1\left(k\right):k]=3$, since $\varrho =1$ iff ${\mathrm{Cl}}_3\left(k\right)\simeq \mathbb{Z}/3\mathbb{Z}$ is elementary cyclic. If $t=2$ and $q_1\leftrightarrow q_2$, then $\varrho =2$, ${\mathrm{Cl}}_3\left(k\right)$ is bicyclic, but may be non-elementary (singular).

In 1995, Ayadi [2] proved that there are only two possibilities for the Galois group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k\right)/k)$ of the 3-class field tower of k with length ${\mathsf{\ell }}_3\left(k\right)$, when $t=2$, $q_1\leftrightarrow q_2$, and ${\mathrm{Cl}}_3\left(k\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is elementary bicyclic (regular), namely, in the notation of [3], either $\mathfrak{G}\simeq \mathrm{SmallGroup}(9,2)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is abelian or $\mathfrak{G}\simeq \mathrm{SmallGroup}(27,4)$ is the extra special 3-group with exponent 9.

The impact of t and $q_1,\dots ,q_t$ on the tower group $\mathfrak{G}$ and its metabelianization $\mathfrak{M}=\mathfrak{G}/{\mathfrak{G}}^{″}$, i.e., the group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ of the second Hilbert 3-class field ${\mathrm{F}}_3^2\left(k\right)$ of k, is shown in

Table 1. Known and unknown impact of t and $q_1,\dots ,q_t$ on $\varrho \left(k\right)$ and $\mathfrak{M}$, $\mathfrak{G}$.

t	Conditions	$\mathit{\varrho }\left(\mathit{k}\right)$	${\mathbf{F}}_3^{\mathit{\infty }}\left(\mathit{k}\right)$	$\mathfrak{M}$	$\mathfrak{G}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
$t=1$		$\varrho =0$	$=k$	$=1$	$=1$	$=0$
$t=2$	$q_1\nleftrightarrow q_2$	$\varrho =1$	$={\mathrm{F}}_3^1\left(k\right)$	$=\mathbb{Z}/3\mathbb{Z}$	$=\mathfrak{M}$	$=1$
$t=2$	$q_1\leftrightarrow q_2$, ${\mathrm{Cl}}_3\left(k\right)$ elem.	$\varrho =2$	$={\mathrm{F}}_3^1\left(k\right)$	$=\mathrm{SmallGroup}(9,2)$	$=\mathfrak{M}$	$=1$
		or	$={\mathrm{F}}_3^2\left(k\right)$	$=\mathrm{SmallGroup}(27,4)$	$=\mathfrak{M}$	$=2$
$t=2$	$q_1\leftrightarrow q_2$, ${\mathrm{Cl}}_3\left(k\right)$ non-elem.	$\varrho =2$	$\ge {\mathrm{F}}_3^2\left(k\right)$	unknown	unknown	$\ge 2$
$t=3$		$2\le \varrho \le 4$	$\ge {\mathrm{F}}_3^1\left(k\right)$	?	?	$\ge 1$

.

However, according to Gras [1], $\varrho =2$ is also possible for $t=3$, and according to Ayadi [2], $\varrho =2$ iff ${\mathrm{Cl}}_3\left(k\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is elementary bicyclic, when $t=3$.

For this situation, $t=3$, $c=pqr$, $\varrho =2$, ${\mathrm{Cl}}_3\left(k\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$, the present article identifies the Galois group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ in dependence on the cubic residue symbols between $p,q,r$. The crucial techniques are based on the lucky coincidence that the four unramified cyclic extensions of degree $[E_i:k]=3$, $1\le i\le 4$, can always be found among the 13 bicyclic bicubic subfields $B_1,\dots ,B_{13}$ of the absolute 3-genus-field $k^{*}$ of k, for which Parry [4] has established a useful class number relation and a structure theory of the unit group. With the aid of the relation rank $d_2\left(\mathfrak{M}\right)\le 4$ or $d_2\left(\mathfrak{M}\right)\ge 5$, it is decided whether $\mathfrak{M}$ coincides with the tower group $\mathfrak{G}$ or not.

The examination of cyclic cubic fields k with $\varrho =3$ and elementary tricyclic ${\mathrm{Cl}}_3\left(k\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^3$ is reserved for a future paper, since among the 13 unramified cyclic extensions of degree $[E_i:k]=3$, $1\le i\le 13$, only 4 are bicyclic bicubic, and the remaining 9 $E_i$ arise in three triplets of pairwise isomorphic non-Galois nonic fields. Similarly, non-elementary ${\mathrm{Cl}}_3\left(k\right)$ for $t=2$ and $p\leftrightarrow q$ is reserved for future investigations.

The present work illuminates Ayadi’s doctoral thesis [2] from the perspective of group theory, and completely clarifies the question mark “?” for the group $\mathfrak{M}$ in the last row of Table 1, partially also the “?” for the group $\mathfrak{G}$, provided that ${\mathrm{Cl}}_3\left(k\right)\simeq {(\mathbb{Z}/3\mathbb{Z})}^2$ is elementary bicyclic.

2. Construction of Cyclic Fields of Odd Prime Degree

2.1. Multiplicity of Conductors and Discriminants

For a fixed odd prime number $\mathsf{\ell }\ge 3$, let k be a cyclic number field of degree ℓ, that is, $k/\mathbb{Q}$ is a Galois extension of degree $[k:\mathbb{Q}]=\mathsf{\ell }$ with absolute automorphism group $\mathrm{Gal}(k/\mathbb{Q})=\langle \sigma \mid {\sigma }^{\mathsf{\ell }}=1\rangle$. According to the Theorem of Kronecker, Weber and Hilbert on abelian extensions of the rational number field $\mathbb{Q}$, the conductor c of k is the smallest positive integer such that $k=k_c$ is contained in the cyclotomic field $K=\mathbb{Q}\left({\zeta }_c\right)$, where ${\zeta }_c=exp(2\pi \sqrt{-1}/c)$ denotes a primitive c-th root of unity, more precisely, in the ℓ-ray class field moduloc of $\mathbb{Q}$, denoted by ${\mathrm{F}}_{\mathsf{\ell },c}\left(\mathbb{Q}\right)$, which lies in the maximal real subfield $K^{+}=\mathbb{Q}({\zeta }_c+{\zeta }_c^{-1})$ of $K=\mathbb{Q}\left({\zeta }_c\right)$.

Theorem 1.

The  conductor   of a cyclic field of odd prime degree ℓ has the shape $c={\mathsf{\ell }}^e·q_1\cdots q_{\tau }$, where $e\in \{0,2\}$ and the $q_i$ are pairwise distinct prime numbers $q_i\equiv +1(\mathrm{mod}\mathsf{\ell })$, for $1\le i\le \tau$. The  discriminant   of $k=k_c$ is the perfect $(\mathsf{\ell }-1)$-th power $d_k=c^{\mathsf{\ell }-1}$, and the number of rational primes which are (totally) ramified in k is given by

$$t:=\left\{\begin{array}{cc}\tau \hfill & \mathit{if}e=0(\mathsf{\ell }\mathit{is}\mathit{unramified}\mathit{in}k),\hfill \\ \tau +1\hfill & \mathit{if}e=2(\mathsf{\ell }\mathit{is}\mathit{ramified}\mathit{in}k).\hfill \end{array}\right.$$

(1)

In the last case, we formally put $q_{\tau +1}:={\mathsf{\ell }}^2$. The number of non-isomorphic cyclic number fields $k_{c,1},\dots ,k_{c,m}$ of degree ℓ, sharing the common conductor c, is given by the  multiplicity formula

$$m=m\left(c\right)={(\mathsf{\ell }-1)}^{t-1}.$$

(2)

Proof. 

See [5] (p. 831). □

2.2. Construction as Ray Class Fields

For the construction of all cyclic number fields $k=k_c$ of degree ℓ with ascending conductors $b\le c\le B$ between an assigned lower bound b and upper bound B by means of the computational algebra system Magma [6], the class field theoretic routines by Fieker [7] can be used without the need to prepare a list of suitable generating polynomials of the ℓ-th degree. The big advantage of this technique is that the cyclic number fields of degree ℓ are produced as a multiplet $(k_{c,1},\dots ,k_{c,m})$ of pairwise non-isomorphic fields sharing the common conductor c with multiplicity $m\in \{1,\mathsf{\ell }-1,{(\mathsf{\ell }-1)}^2,{(\mathsf{\ell }-1)}^3,\dots \}$ in dependence on the number $t\in \{1,2,3,4,\dots \}$ of primes dividing the conductor c, according to Formula (2). Our algorithms for the construction, and statistics of ℓ-class groups, have been presented in [8] (Alg. 1–3, pp. 4–7, Tbl. 1.1–1.6, pp. 7–11). From now on, let $\mathsf{\ell }=3$ for the remainder of this article.

3. Arithmetic of Cyclic Cubic Fields

Generally, t denotes the number of prime divisors of the conductor c of a cyclic cubic number field k, and $\varrho \left(k\right)={\varrho }_3\left(k\right)$ denotes the rank ${dim}_{{\mathbb{F}}_3}({\mathrm{Cl}}_3\left(k\right)/{\mathrm{Cl}}_3{\left(k\right)}^3)$ of the 3-class group ${\mathrm{Cl}}_3\left(k\right)={\mathrm{Syl}}_3\mathrm{Cl}\left(k\right)$. In formulas concerning principal factors (Section 3.2), the prime power conductor $3^2$ must be replaced by 3.

3.1. Rank of 3-Class Groups

Since the rank ${\varrho }_3\left(k\right)$ of the 3-class group ${\mathrm{Cl}}_3\left(k\right)$ of a cyclic cubic field k depends on the mutual cubic residue conditions between the prime(power) divisors $q_1,\dots ,q_t$ of the conductor c, Gras [1] (pp. 21–22) has introduced directed graphs with t vertices $q_1,\dots ,q_t$ whose directed edges $q_i\to q_j$ describe values of cubic residue symbols. We use a simplified notation of these graphs, fitting in a single line, but occasionally requiring the repetition of a vertex.

Definition 1.

Let ${\zeta }_3$ be a fixed primitive third root of unity. For each pair $(q_i,q_j)$ with $1\le i\ne j\le t$, the value of the cubic residue symbol ${\left(\frac{q_i}{q_j}\right)}_3={\zeta }_3^{a_{ij}}$ is determined uniquely by the integer $a_{ij}\in \{-1,0,1\}$. Let a directed edge $q_i\to q_j$ be defined if and only if ${\left(\frac{q_i}{q_j}\right)}_3=1$, that is, $q_i$ is a cubic residue modulo $q_j$ (and thus, $a_{ij}=0$). The  combined cubic residue symbol   ${[q_1,\dots ,q_t]}_3:=$

$$\left\{q_i\to q_j|i\ne j,{\left(\frac{q_i}{q_j}\right)}_3=1\right\}\bigcup \left\{q_i|(\forall j\ne i){\left(\frac{q_i}{q_j}\right)}_3\ne 1,{\left(\frac{q_j}{q_i}\right)}_3\ne 1\right\}$$

(3)

where the subscripts i and j run from 1 to t is defined as the union of the set of all directed edges that occur in the graph associated with $q_1,\dots ,q_t$ in the sense of Gras, and the set of all isolated vertices. For $t=3$, we additionally need the invariant $\delta :=a_{12}{a}_{23}{a}_{31}-a_{13}{a}_{32}{a}_{21}$ in order to distinguish two subcases of the case with three isolated vertices.

Theorem 2

(Rank Distribution, G. Gras, 1973, [1] (Prop. VI.5, pp. 21–22)). Let k be a cyclic cubic field of conductor $c=q_1\cdots q_t$ with $1\le t\le 3$. We indicate mutual cubic residues simply by writing $q_1\leftrightarrow q_2$ instead of $q_1\to q_2\to q_1$.

• If $t=1$, then $m=1$, k forms a singlet, ${\left[q_1\right]}_3=\left\{q_1\right\}$, and $\varrho \left(k\right)=0$.
• If $t=2$, then $m=2$, k is member of a doublet $(k_1,k_2)$, and there arise two possibilities.

  1.

  $(\varrho \left(k_1\right),\varrho \left(k_2\right))=(1,1)$, if

  $${[q_1,q_2]}_3=\left\{\begin{array}{cc}\{q_1,q_2\}\hfill & ,\mathit{Graph}\mathbf{1},\mathit{or}\hfill \\ \{q_i\to q_j\}\hfill & ,\mathit{Graph}\mathbf{2},\mathit{with}i\ne j.\hfill \end{array}\right.$$

  (4)

  2.

  $(\varrho \left(k_1\right),\varrho \left(k_2\right))=(2,2)$, if

  $${[q_1,q_2]}_3=\{q_1\leftrightarrow q_2\},\mathit{Graph}\mathbf{3}.$$

  (5)

• If $t=3$, then $m=4$, k is member of a quartet $(k_1,\dots ,k_4)$, and there arise five cases.

  1.

  $(\varrho \left(k_1\right),\dots ,\varrho \left(k_4\right))=(2,2,2,2)$, called  Category $\mathbf{III}$, if

  $${[q_1,q_2,q_3]}_3=\left\{\begin{array}{cc}\{q_1,q_2,q_3;\delta \not\equiv 0\left(\mathrm{mod}3\right)\}\hfill & ,\mathit{Graph}\mathbf{1},\mathit{or}\hfill \\ \{q_i\to q_j;q_l\}\hfill & ,\mathit{Graph}\mathbf{2},\mathit{or}\hfill \\ \{q_i\to q_j\to q_l\}\hfill & ,\mathit{Graph}\mathbf{3},\mathit{or}\hfill \\ \{q_i\to q_j\to q_l\to q_i\}\hfill & ,\mathit{Graph}\mathbf{4},\mathit{or}\hfill \\ \{q_i\leftrightarrow q_j;q_l\}\hfill & ,\mathit{Graph}\mathbf{5},\mathit{or}\hfill \\ \{q_i\leftrightarrow q_j\to q_l\}\hfill & ,\mathit{Graph}\mathbf{6},\mathit{or}\hfill \\ \{q_i\leftrightarrow q_j\leftarrow q_l\}\hfill & ,\mathit{Graph}\mathbf{7},\mathit{or}\hfill \\ \{q_l\to q_i\leftrightarrow q_j\leftarrow q_l\}\hfill & ,\mathit{Graph}\mathbf{8},\mathit{or}\hfill \\ \{q_l\to q_i\leftrightarrow q_j\to q_l\}\hfill & ,\mathit{Graph}\mathbf{9}\hfill \end{array}\right.$$

  (6)

  with $i,j,l$ pairwise distinct.

  2.

  $(\varrho \left(k_1\right),\dots ,\varrho \left(k_4\right))=(3,2,2,2)$, called  Category $\mathbf{I}$, if

  $${[q_1,q_2,q_3]}_3=\left\{\begin{array}{cc}\{q_1,q_2,q_3;\delta \equiv 0\left(\mathrm{mod}3\right)\}\hfill & ,\mathit{Graph}\mathbf{1},\mathit{or}\hfill \\ \{q_i\leftarrow q_j\to q_l\}\hfill & ,\mathit{Graph}\mathbf{2}\hfill \end{array}\right.$$

  (7)

  with $i,j,l$ pairwise distinct.

  3.

  $(\varrho \left(k_1\right),\dots ,\varrho \left(k_4\right))=(3,3,2,2)$, called  Category $\mathbf{II}$, if

  $${[q_1,q_2,q_3]}_3=\left\{\begin{array}{cc}\{q_i\to q_j\leftarrow q_l\}\hfill & ,\mathit{Graph}\mathbf{1},\mathit{or}\hfill \\ \{q_i\to q_j\leftarrow q_l\to q_i\}\hfill & ,\mathit{Graph}\mathbf{2}\hfill \end{array}\right.$$

  (8)

  with $i,j,l$ pairwise distinct.

  4.

  $(\varrho \left(k_1\right),\dots ,\varrho \left(k_4\right))=(3,3,3,3)$, called  Category $\mathbf{IV}$, if

  $${[q_1,q_2,q_3]}_3=\left\{\begin{array}{cc}\{q_i\leftarrow q_j\leftrightarrow q_l\to q_i\}\hfill & ,\mathit{Graph}\mathbf{1},\mathit{or}\hfill \\ \{q_i\leftrightarrow q_j\leftrightarrow q_l\}\hfill & ,\mathit{Graph}\mathbf{2},\mathit{or}\hfill \\ \{q_i\leftrightarrow q_j\leftrightarrow q_l\to q_i\}\hfill & ,\mathit{Graph}\mathbf{3}\hfill \end{array}\right.$$

  (9)

  with $i,j,l$ pairwise distinct.

  5.

  $(\varrho \left(k_1\right),\dots ,\varrho \left(k_4\right))=(4,4,4,4)$, called  Category $\mathbf{V}$, if

  $${[q_1,q_2,q_3]}_3=\{q_1\leftrightarrow q_2\leftrightarrow q_3\leftrightarrow q_1\}.$$

  (10)

Proof. 

See [1] (Prp. VI.5, pp. 21–22). Multiplicities $m\in \{1,2,4\}$ are taken from Theorem 1. □

Remark 1.

Ayadi introduced categories in [2] (pp. 45–47). He investigated the cases $t=2$, Formula (5); and $t=3$, Formulas (6)–(8), in Theorem 2. For $t=3$, he denoted the nine subcases of Formula (6) by Graph 1,2,3,4,5,6,7,8,9 of Category III, the two subcases of Formula (7) by Graph 1,2 of Category I, and the two subcases of Formula (8) by Graph 1,2 of Category II. For Categories I and II, Ayadi did  not   study the fields with 3-class rank ${\varrho }_3\left(k_{\mu }\right)=3$, $1\le \mu \le 4$. Our algorithms for the classification by categories and graphs, and their statistics, have been presented in [8] (Alg. 4–5, Tbl. 2.1, pp. 15–19).

For $t=3$, we also write briefly $p=q_1$, $q=q_2$ and $r=q_3$ for the prime(power)s dividing the conductor $c=pqr$.

• Graph 1 of Category $\mathrm{I}$ with symbol ${[p,q,r]}_3=\{p,q,r;\delta \equiv 0\left(\mathrm{mod}3\right)\}$ and
• Graph 1 of Category $\mathrm{III}$ with symbol ${[p,q,r]}_3=\{p,q,r;\delta \not\equiv 0\left(\mathrm{mod}3\right)\}$ are the only two situations without any trivial cubic residue conditions between $p,q,r$.
• We show the impact of the $\delta$-invariant.

Lemma 1.

Consider three cubic residue symbols for products of two primes, ${\left(\frac{qr}{p}\right)}_3$, ${\left(\frac{pr}{q}\right)}_3$, ${\left(\frac{pq}{r}\right)}_3$ with respect to triviality, i.e., being equal to 1.

If $\delta \equiv 0$, then zero or two of the symbols are trivial.

If $\delta \not\equiv 0$, then one or three of the symbols are trivial.

Proof. 

For each of the two triplets $(a_{12},a_{23},a_{31})$ and $(a_{32},a_{13},a_{21})$ of exponents in Definition 1, there are $2^3=8$ combinatorial possibilities. The product of the components is $+1$ if zero or two components are negative, and it is $-1$ if one or three components are negative.

For Graph $\mathrm{I}.1$ with $\delta \equiv 0$, triplets with equal product must be combined. Consequently, for each choice of a fixed first triplet, one of the four admissible second triplets (namely $(a_{32},a_{13},a_{21})=(a_{12},a_{23},a_{31})$) produces no trivial symbol, and three of the second triplets produce two trivial symbols each.

For Graph $\mathrm{III}.1$ with $\delta \not\equiv 0$, triplets with distinct products must be combined. Consequently, for each choice of a fixed first triplet, one of the four admissible second triplets (namely $(a_{32},a_{13},a_{21})=-(a_{12},a_{23},a_{31})$) produces three trivial symbols, and three of the second triplets produce a single trivial symbol each. □

3.2. Ambiguous Principal Ideals

The number of primitive ambiguous ideals of a cyclic cubic field k, which are invariant under $\mathrm{Gal}(k/\mathbb{Q})=\langle \sigma \rangle$, increases with the number t of prime factors of the conductor c. According to Hilbert’s Theorem 93, the number is given by

$$\#\left({\mathcal{I}}_k^{\langle \sigma \rangle }/{\mathcal{I}}_{\mathbb{Q}}\right)=3^t.$$

(11)

However, the number of primitive ambiguous principal ideals of k is a fixed invariant of all cyclic cubic fields, regardless of the number t.

Theorem 3.

The number of ambiguous principal ideals of any cyclic cubic field k is given by

$$\#\left({\mathcal{P}}_k^{\langle \sigma \rangle }/{\mathcal{P}}_{\mathbb{Q}}\right)=3.$$

(12)

Proof. 

The well-known theorem on the Herbrand quotient of the unit group $U_k$ of k as a Galois module over the group $\mathrm{Gal}(k/\mathbb{Q})=\langle \sigma \rangle$, which can be expressed by abstract cohomology groups $\#{\mathrm{H}}^{-1}(\langle \sigma \rangle ,U_k)/\#{\widehat{\mathrm{H}}}^0(\langle \sigma \rangle ,U_k)=[k:\mathbb{Q}]$, can also be stated more ostensively as $\#\left({\mathcal{P}}_k^{\langle \sigma \rangle }/{\mathcal{P}}_{\mathbb{Q}}\right)=\#\left(E_{k/\mathbb{Q}}/U_k^{1-\sigma }\right)=[k:\mathbb{Q}]·\#\left(U_{\mathbb{Q}}/N_{k/\mathbb{Q}}\left(U_k\right)\right)=3$, since the unit norm index is given by $\left(U_{\mathbb{Q}}:N_{k/\mathbb{Q}}\left(U_k\right)\right)=1$. Here, $E_{k/\mathbb{Q}}=\{\epsilon \in U_k\mid N_{k/\mathbb{Q}}\left(\epsilon \right)=1\}$ are the relative units. □

Consequently, if we speak about a non-trivial primitive ambiguous principal ideal of k, then we either mean $\left(\alpha \right)=\alpha {\mathcal{O}}_k$ or $({\alpha }^2/b)=({\alpha }^2/b){\mathcal{O}}_k$, where ${\mathcal{P}}_k^{\langle \sigma \rangle }/{\mathcal{P}}_{\mathbb{Q}}=\{1,\left(\alpha \right),({\alpha }^2/b)\}$. The norms of these two elements are divisors of the square $c^2=q_1^2\cdots q_t^2$ of the conductor c of k, where $q_t$ must be replaced by 3 if $q_t=9$. When $N_{k/\mathbb{Q}}\left(\alpha \right)=a·b^2$ with square-free coprime integers $a,b$, then $N_{k/\mathbb{Q}}({\alpha }^2/b)=a^2·b^4/b^3=a^2·b$. It follows that both norms are cube-free integers.

Definition 2.

The minimum of the two norms of non-trivial primitive ambiguous principal ideals $\left(\alpha \right),({\alpha }^2/b)$ of a cyclic cubic field k is called the  principal factor   (of the discriminant $d_k=c^2$) of the field k, denoted by $A\left(k\right):=min\{a·b^2,a^2·b\}$, that is,

$$A\left(k\right)=\left\{\begin{array}{cc}a·b^2\hfill & \mathit{if}b<a,\hfill \\ a^2·b\hfill & \mathit{if}a<b.\hfill \end{array}\right.$$

(13)

Ayadi [2,9] (Rem. 2.6, p. 18) speaks about the Parry constant or Parry invariant $A\left(k\right)$ of k, and Derhem [10] calls $A\left(k\right)=N_{k/\mathbb{Q}}\left(R\right)$ with $R=1+\epsilon +{\epsilon }^{1+\sigma }$, $\epsilon =R^{1-\sigma }$, the Kummer resolvent of k, when $U_k=\langle -1,\epsilon ,{\epsilon }^{\sigma }\rangle$ as a $\langle \sigma \rangle$-module is generated by $-1$ and the fundamental unit $\epsilon$. However, the concept of principal factors has been coined much earlier by Barrucand and Cohn [11]. Our algorithm for the determination of principal factors has been presented in [8] (Alg. 6, pp. 20–21).

Theorem 4

(Principal factor criterion, Ayadi, 1995, [2] (Thm. 3.3, p. 37)). Let c be a conductor divisible by two primes, $t=2$, such that ${\mathrm{Cl}}_3\left(k_{c,\mu }\right)\simeq (3,3)$ for both cyclic cubic fields $k_{c,\mu }$, $1\le \mu \le 2$, with conductor c. Denote by $\mathcal{P}$ the number of prime divisors of the norm $A\left(k\right)={\mathrm{N}}_{k/\mathbb{Q}}\left(\alpha \right)$ of a non-trivial primitive ambiguous principal ideal $\left(\alpha \right)$, i.e., a principal factor, of any of the two fields $k=k_{c,\mu }$. Then, $\mathcal{P}\in \{1,2\}$, and the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ of both fields $k=k_{c,\mu }$ is given by

$$\mathfrak{M}\simeq \left\{\begin{array}{cc}\langle 9,2\rangle \mathit{with}\mathit{capitulation}\mathit{type}\mathrm{a}.1,\varkappa \left(k\right)=\left(0000\right),\hfill & \mathit{if}\mathcal{P}=2,\hfill \\ \langle 27,4\rangle \mathit{with}\mathit{capitulation}\mathit{type}\mathrm{A}.1,\varkappa \left(k\right)=\left(1111\right),\hfill & \mathit{if}\mathcal{P}=1.\hfill \end{array}\right.$$

(14)

The length of the Hilbert 3-class field tower is ${\mathsf{\ell }}_3\left(k\right)=1$ with ${\mathrm{F}}_3^{\infty }\left(k\right)={\mathrm{F}}_3^1\left(k\right)$ if $\mathcal{P}=2$, and ${\mathsf{\ell }}_3\left(k\right)=2$ with ${\mathrm{F}}_3^{\infty }\left(k\right)={\mathrm{F}}_3^2\left(k\right)$ if $\mathcal{P}=1$. In both cases, $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k\right)/k)=\mathfrak{M}$.

Proof. 

See [2] (Prp. 3.6, p. 32, Thm. 3.1, p. 34, Thm. 3.3, p. 37) and [8] (pp. 31–33). □

The first example $c=19·1129=21451$ for $\mathfrak{M}\simeq \langle 27,4\rangle$ is due to Scholz and Taussky [12] (pp. 209–210). It was misprinted as $19·1429=27151$ in [13] (p. 383). Systematic tables are presented at http://www.algebra.at/ResearchFrontier2013ThreeByThree.htm (accessed on 10 October 2023) in Sections 1.1 and 1.2.

Concerning the 3-capitulation types $\mathrm{a}.1$ and $\mathrm{A}.1$, viewed as transfer kernel types (TKT), and the related concept of transfer target types (TTT), i.e., abelian-type invariants (ATI), see [14].

4. Unramified Extensions of Cyclic Cubic Fields

In this crucial section, we first introduce the absolute 3-genus field $k^{*}$ (Section 4.1) of a cyclic cubic number field k. Then we show that the bicyclic bicubic subfields $B<k^{*}$ constitute unramified cyclic cubic relative extensions $B/k$ of a cyclic cubic number field k with $t=3$. Finally, using the unramified cyclic cubic relative extensions $E/k$ as capitulation targets (Section 4.3), we define the capitulation kernels (Section 4.2) of a cyclic cubic number field k with the non-trivial 3-class group ${\mathrm{Cl}}_3\left(k\right)$.

4.1. The Absolute 3-Genus Field

The absolute 3-genus field $k^{*}={(k/\mathbb{Q})}^{*}$ of a cyclic cubic field k is the maximal unramified 3-extension $k^{*}/k$ with the abelian absolute Galois group $\mathrm{Gal}(k^{*}/\mathbb{Q})$. If the conductor $c=q_1\cdots q_t$ of $k=k_c$ has t prime divisors, then $k^{*}$ is the compositum of the multiplet $(k_{c,1},\dots ,k_{c,m})$ of all cyclic cubic fields sharing the common conductor c, where $m=m\left(c\right)=2^{t-1}$, according to the multiplicity Formula (2). The absolute Galois group $\mathrm{Gal}(k^{*}/\mathbb{Q})$ is the elementary abelian 3-group ${(\mathbb{Z}/3\mathbb{Z})}^t$. In particular, if $t=1$, $c=q_1$, then $k^{*}=k$ is the cyclic cubic field itself, and if $t=2$, $c=q_1{q}_2$, then $k^{*}=k_{c,1}·k_{c,2}$ is a bicyclic bicubic field with conductor c and discriminant

$$d\left(k^{*}\right)=d\left(k_{q_1}\right)·d\left(k_{q_2}\right)·d\left(k_{c,1}\right)·d\left(k_{c,2}\right)=q_1^2·q_2^2·{\left(q_1{q}_2\right)}^2·{\left(q_1{q}_2\right)}^2=c^6.$$

(15)

In 1990, Parry [4] investigated the arithmetic of a general bicyclic bicubic field $B/\mathbb{Q}$ with conductor $c=q_1\cdots q_t$, $t\ge 2$, and four cyclic cubic subfields $k_1,\dots ,k_4$. In particular, he determined the class number relation in terms of the index I of subfield units of B.

Theorem 5.

Let $M:=\left(e_{i,j}\right)$ be the $(4\times t)$-matrix of   integer exponents in the following representation of The  principal factors $A\left(k_i\right)={\prod }_{j=1}^t{q}_j^{e_{i,j}}$, for $1\le i\le 4$. Then:

1.

The Galois group $\mathrm{Gal}(B/\mathbb{Q})\simeq (3,3)$ is elementary bicyclic.

2.

The index $I:=(U:V)$ of the subgroup $V:=\langle U_1,\dots ,U_4\rangle$ generated by the unit groups $U_i:=U_{k_i}$, $1\le i\le 4$, in the unit group $U:=U_B$ is bounded by $I=3^e$, $0\le e\le 3$.

3.

The  class number   of B satisfies the following  relation:

$$h\left(B\right)=\frac{I}{3^5}·\prod _{i=1}^{4}h\left(k_i\right)=\frac{(U:V)}{243}·h\left(k_1\right)·h\left(k_2\right)·h\left(k_3\right)·h\left(k_4\right),$$

(16)

where I denotes the abovementioned  index of subfield units   of B.

4.

$3\nmid h\left(B\right)$ if and only if $c=pq$, i.e., $t=2$, and $p\nleftrightarrow q$ are  not   mutual cubic residues, i.e., the graph of $p,q$ is either Graph 1 or Graph 2. If $3\nmid h\left(B\right)$, then $I=27$.

5.

In dependence on the rank $2\le r_M:=\mathrm{rank}\left(M\right)\le 4$ of the matrix M, the  index  I takes the following values:

$$I=(U:V)=\left\{\begin{array}{cc}1\hfill & \mathit{if}{r}_M=4,\hfill \\ 3\hfill & \mathit{if}{r}_M=3,\hfill \\ 9\mathit{or}27\hfill & \mathit{if}{r}_M=2.\hfill \end{array}\right.$$

(17)

Proof. 

For the class number relation, see Parry [4] (Prp. 7, p. 496, Thm. 9, p. 497). Generally, the index of subfield units, I, is a divisor of $27=3^3$ [4] (Lem. 11, p. 500, Thm. 13, p. 501). See also Ayadi [2] (Prop. 2.7.(2) and Prop. 2.8, p. 20). Note that $p\nleftrightarrow q$ implies ${\prod }_{i=1}^{4}h\left(k_i\right)=9$. □

Corollary 1.

Let $t=3$ and B be a bicyclic bicubic field with conductor $c=pqr$ such that there are  no mutual cubic residues   among $p,q,r$. Then:

1.

For all $1\le j\le 4$, $h_3\left(B_j\right)=\frac{(U_j:V_j)}{3^2}{h}_3\left(k_j\right)$.

2.

For all $5\le j\le 10$, $h_3\left(B_j\right)=\frac{(U_j:V_j)}{3^4}{h}_3\left(k_i\right)h_3\left(k_{\mathsf{\ell }}\right)$, where $1\le i,\mathsf{\ell }\le 4$, $i\ne \mathsf{\ell }$, and $k_i$, $k_{\mathsf{\ell }}$ are the two components of the quartet which are contained in $B_j$.

Proof. 

By (22), the first statement is valid, since $h_3\left(k\right)=3$ for the six subfields k with $t=2$. By (23), the second statement holds, since $h_3\left(k\right)=1$ for the three subfields k with $t=1$. □

For a cyclic cubic field k with $t=2$, $c=pq$, the 3-class numbers of the 3-genus field $k^{*}$, which is bicyclic bicubic, and of its four cyclic cubic subfields can be summarized as follows.

Theorem 6.

Let $k^{*}=k_p·k_q·k_{c,1}·k_{c,2}$ be the genus field of the two cyclic cubic fields $k_{c,1}$ and $k_{c,2}$ with conductor $c=pq$. Denote the 3-valuations of the class numbers $h^{*}$, $h_1$, $h_2$, $h_3$, $h_4$ of $k^{*}$, $k_p$, $k_q$, $k_{c,1}$, $k_{c,2}$, respectively, by $v^{*}$, $v_1$, $v_2$, $v_3$, $v_4$. Then, $v_1=v_2=0$, and

$$v^{*}\left\{\begin{array}{cc}=0,v_3=v_4=1,I=27,\hfill & \mathit{if}p\nleftrightarrow q,\hfill \\ =1,\hfill & \mathit{if}p\leftrightarrow q,v_3=v_4=2,I=9,\hfill \\ =2,\hfill & \mathit{if}p\leftrightarrow q,v_3=v_4=2,I=27,\hfill \\ \ge 3,\hfill & \mathit{if}p\leftrightarrow q,v_3\ge 3,v_4\ge 3,I\ge 9.\hfill \end{array}\right.$$

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Proof. 

According to Theorem 2, we generally have $v_1=v_2=0$, $v_3\ge 1$, $v_4\ge 1$ if $p\nleftrightarrow q$, and $v_3\ge 2$, $v_4\ge 2$ if $p\leftrightarrow q$. Now, the claim is a consequence of Formula (16), which yields

$$v^{*}=v_3\left(h^{*}\right)=v_3\left(I\right)-5+\sum _{i=1}^4{v}_3\left(h_i\right)=v_3\left(I\right)-5+v_1+v_2+v_3+v_4=v_3\left(I\right)-5+v_3+v_4.$$

The combination of [4] (Thm. 9, p. 497 and Cor. 1, p. 498) shows that $v^{*}=0$ if and only if $p\nleftrightarrow q$, and $v^{*}=0$ implies $v_3\left(I\right)=3$, whence necessarily $v_3=v_4=1$. However, if $p\leftrightarrow q$, then $v_3=2$ is equivalent with $v_4=2$, according to [9] (Thm. 4.1, p. 472). □

Remark 2.

For $v_3=v_4=2$, we have ${\mathrm{Cl}}_3\left(k_{pq,\mu }\right)\simeq (3,3)$. The smallest occurrences of $v_3=v_4=3$ are the conductors $7·673=4711$ (“Eau de Cologne”,  singular  with ${\mathrm{Cl}}_3\left(k^{*}\right)\simeq (3,3,3)$) and $7·769=5383$ (super-singular   with ${\mathrm{Cl}}_3\left(k^{*}\right)\simeq (9,3,3)$) both with ${\mathrm{Cl}}_3\left(k_{pq,\mu }\right)\simeq (9,3)$, for $\mu \in \{1,2\}$.

For a cyclic cubic field k with $t=3$ and conductor $c=q_1{q}_2{q}_3$, the 3-genus field $k^{*}$ contains 13 bicyclic bicubic subfields. Three of them are the sub-genus fields $B_i:={\left(k_{f_{i-10}}\right)}^{*}$, $11\le i\le 13$, of the cyclic cubic fields with conductors $f_1=q_1{q}_2$, $f_2=q_1{q}_3$, $f_3=q_2{q}_3$, respectively. In the numerical tables of [8], we always start with the leading three sub-genus fields $B_i$, $11\le i\le 13$, separated by a semicolon from the trailing ten remaining bicyclic bicubic subfields, when we give a family of invariants for these 13 subfields $B_1,\dots ,B_{13}$,

$$\mathrm{in}\mathrm{particular},{\left[{\mathrm{Cl}}_3{B}_i\right]}_{1\le i\le 13}:=[{\mathrm{Cl}}_3\left(B_{11}\right),\dots ,{\mathrm{Cl}}_3\left(B_{13}\right);{\mathrm{Cl}}_3\left(B_1\right),\dots ,{\mathrm{Cl}}_3\left(B_{10}\right)].$$

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4.2. Capitulation Kernels

We recall the connection between the size of the capitulation kernel $ker\left(T_{E/k}\right)$ and the unit norm index $(U_k:{\mathrm{N}}_{E/k}\left(U_E\right))$ of an unramified cyclic cubic extension $E/k$ of a cyclic cubic field k. Here, $T_{E/k}:{\mathrm{Cl}}_3\left(k\right)\to {\mathrm{Cl}}_3\left(E\right)$, $\mathfrak{a}{\mathcal{P}}_k\mapsto \left(\mathfrak{a}{\mathcal{O}}_E\right){\mathcal{P}}_E$, denotes the extension homomorphism or transfer of 3-classes from k to E.

Theorem 7.

The order of the 3-capitulation kernel or  transfer kernel   of $E/k$ is given by

$$\#ker\left(T_{E/k}\right)=\left\{\begin{array}{c}3,\hfill \\ 9,\hfill \\ 27,\hfill \end{array}\right.\mathit{if}\mathit{and}\mathit{only}\mathit{if}(U_k:{\mathrm{N}}_{E/k}\left(U_E\right))=\left\{\begin{array}{c}1,\hfill \\ 3,\hfill \\ 9.\hfill \end{array}\right.$$

(20)

Proof. 

According to the Herbrand Theorem on the cohomology of the unit group $U_E$ as a Galois module with respect to $G=\mathrm{Gal}(E/k)$, we have the relation $\#ker\left(T_{E/k}\right)=[E:k]·(U_k:{\mathrm{N}}_{E/k}\left(U_E\right))$, since $ker\left(T_{E/k}\right)\simeq H^1(G,U_E)$ when $E/k$ is unramified of odd prime degree $[E:k]=3$ and $U_k/{\mathrm{N}}_{E/k}\left(U_E\right)\simeq {\widehat{H}}^0(G,U_E)$. The cyclic cubic base field k has the signature $(r_1,r_2)=(3,0)$ and torsion-free Dirichlet unit rank $r=r_1+r_2-1=3+0-1=2$. Thus, there are three possibilities for the unit norm index $(U_k:{\mathrm{N}}_{E/k}\left(U_E\right))\in \{1,3,9\}$. □

Remark 3.

When k is a cyclic cubic field with 3-class group $O:={\mathrm{Cl}}_3\left(k\right)$ of elementary tricyclic type $(3,3,3)$, viewed as a vector space of dimension 3 over the finite field ${\mathbb{F}}_3$, then $\#ker\left(T_{E/k}\right)=3$ if and only if $ker\left(T_{E/k}\right)=L_i$ is a  line   for some $1\le i\le 13$, $\#ker\left(T_{E/k}\right)=9$ if and only if $ker\left(T_{E/k}\right)=P_i$ is a  plane   for some $1\le i\le 13$, and $\#ker\left(T_{E/k}\right)=27$ if and only if $ker\left(T_{E/k}\right)=O$ is the  entire vector space   over ${\mathbb{F}}_3$. Details are reserved for a future paper. Our algorithms for the determination of the capitulation kernels for ${\mathrm{Cl}}_3\left(k\right)$ of type $(3,3)$ and $(3,3,3)$ have been presented in [8] (Alg. 8–9, pp. 26–30).

In our theorems on cyclic cubic fields with $t=3$ belonging to the various graphs of each category, we shall frequently find particular statements which relate several similar capitulation types.

Definition 3.

Let G be a 3-group with generator rank $d_1\left(G\right)=2$ and elementary bicyclic commutator quotient $G/G^{\prime }\simeq (3,3)$. By $T_{G,H_i}:G/G^{\prime }\to H_i/H_i^{\prime }$, we denote the transfers from G to the four maximal normal subgroups $H_i$, $1\le i\le 4$. Then, the set of all ordered transfer kernel types $\varkappa ={\left({\varkappa }_i\right)}_{1\le i\le 4}$ with ${\varkappa }_i:=ker\left(T_{G,H_i}\right)$ is endowed with a partial order relation $\varkappa \le {\varkappa }^{\prime }$ by $(\forall 1\le i\le 4)$${\varkappa }_i\le {\varkappa }_i^{\prime }$. The order is strict, $\varkappa <{\varkappa }^{\prime }$, when $\varkappa \le {\varkappa }^{\prime }$ and $(\exists 1\le j\le 4)$${\varkappa }_j<{\varkappa }_j^{\prime }$.

The possibilities for a strict order are rather limited, since a transfer kernel is either cyclic of order 3 (partial—by Hilbert’s Theorem 94, it cannot be trivial) or bicyclic of type $(3,3)$ (total). As usual, we abbreviate ${\varkappa }_i=j$ if $(\exists 1\le j\le 4)$$ker\left(T_{G,H_i}\right)=H_j/G^{\prime }$, and ${\varkappa }_i=0$ if $ker\left(T_{G,H_i}\right)=G/G^{\prime }$, for fixed $1\le i\le 4$. So, $\varkappa <{\varkappa }^{\prime }$⟺ $(\exists 1\le j,i\le 4)$${\varkappa }_j=H_i/G^{\prime }<G/G^{\prime }={\varkappa }_j^{\prime }$. The arithmetical application of this group theoretic Definition 3 is given in the following definition.

Definition 4.

Let K be an algebraic number field with the elementary bicyclic 3-class goup ${\mathrm{Cl}}_3\left(K\right)\simeq (3,3)$. Then, K has four unramified cyclic cubic relative extensions $E_i/K$, $1\le i\le 4$, and corresponding class extension homomorphisms $T_{E_i/K}:{\mathrm{Cl}}_3\left(K\right)\to {\mathrm{Cl}}_3\left(E_i\right)$. Let $\mathfrak{M}:=\mathrm{Gal}({\mathrm{F}}_3^2\left(K\right)/K)$ be the Galois group of the second Hilbert 3-class field of K, that is, the maximal metabelian unramified 3-extension of K. Then, $\varkappa \left(K\right):=\varkappa \left(\mathfrak{M}\right)$ is called the  minimal transfer kernel type   (mTKT) of K, if $\varkappa \left(K\right)\le {\varkappa }^{\prime }\left(K\right)$, for any other possible capitulation type ${\varkappa }^{\prime }\left(K\right)$.

4.3. Capitulation Targets

The precise constitution of the lattice of all subfields of the absolute 3-genus field $k^{*}$ of a cyclic cubic field $k=k_{pqr}$ with $t=3$ and conductor $c=pqr$ is as follows.

Theorem 8.

The genus field $k^{*}$ of k contains 13 cyclic cubic fields:

$$\begin{array}{cc}\hfill & k_{p,1},k_{q,1},k_{r,1},k_{pq,1},k_{pq,2},k_{pr,1},k_{pr,2},k_{qr,1},k_{qr,2},k_{pqr,1},k_{pqr,2},k_{pqr,3},k_{pqr,4},\mathit{briefly}\hfill \\ \hfill & k_p,k_q,k_r,k_{pq},{\tilde{k}}_{pq},k_{pr},{\tilde{k}}_{pr},k_{qr},{\tilde{k}}_{qr},k_1,k_2,k_3,k_4.\hfill \end{array}$$

(21)

• The composita $L:=k_{pq}{k}_{pr}{k}_{qr}$ and $\tilde{L}:={\tilde{k}}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$ satisfy the  skew balance of degrees
• $[L:\mathbb{Q}]·[\tilde{L}:\mathbb{Q}]=243$ with $[L:\mathbb{Q}]=9$ and $[\tilde{L}:\mathbb{Q}]=27$, or vice versa.
• Alert:   Always in the sequel, the  normalization $[L:\mathbb{Q}]=9$ is assumed.
• The genus field $k^{*}$ of k contains 13 bicyclic bicubic fields:

$$\begin{array}{cc}\hfill 4 \mathit{single}-\mathit{capitulation}\mathit{targets}{B}_1& :=k_{pq}{k}_{pr}=k_1{k}_{pq}{k}_{pr}{k}_{qr},\hfill \\ \hfill B_2& :={\tilde{k}}_{pr}{\tilde{k}}_{qr}=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr},\hfill \\ \hfill B_3& :={\tilde{k}}_{pq}{\tilde{k}}_{pr}=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr},\hfill \\ \hfill B_4& :={\tilde{k}}_{pq}{\tilde{k}}_{qr}=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill 6 \mathit{double}-\mathit{capitulation}\mathit{targets}{B}_5& :=k_p{\tilde{k}}_{qr}=k_1{k}_3{k}_p{\tilde{k}}_{qr},\hfill \\ \hfill B_6& :=k_q{\tilde{k}}_{pr}=k_1{k}_4{k}_q{\tilde{k}}_{pr},\hfill \\ \hfill B_7& :=k_r{\tilde{k}}_{pq}=k_1{k}_2{k}_r{\tilde{k}}_{pq},\hfill \\ \hfill B_8& :=k_p{k}_{qr}=k_2{k}_4{k}_p{k}_{qr},\hfill \\ \hfill B_9& :=k_q{k}_{pr}=k_2{k}_3{k}_q{k}_{pr},\hfill \\ \hfill B_{10}& :=k_r{k}_{pq}=k_3{k}_4{k}_r{k}_{pq},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill \mathit{and}3 \mathit{sub}-\mathit{genus}\mathit{fields}{B}_{11}& :=k_{pq}{\tilde{k}}_{pq}=k_p{k}_q{k}_{pq}{\tilde{k}}_{pq},\hfill \\ \hfill B_{12}& :=k_{pr}{\tilde{k}}_{pr}=k_p{k}_r{k}_{pr}{\tilde{k}}_{pr},\hfill \\ \hfill B_{13}& :=k_{qr}{\tilde{k}}_{qr}=k_q{k}_r{k}_{qr}{\tilde{k}}_{qr},\hfill \end{array}$$

(24)

provided that $k_{pq,}{k}_{pr},k_{qr}$ are normalized. The conductor of $B_1,\dots ,B_{10}$ is $c=pqr$, the conductor of $B_{11}$ is $f_1=pq$, the conductor of $B_{12}$ is $f_2=pr$, and the conductor of $B_{13}$ is $f_3=qr$.

Proof. 

See [2] (Prop. 4.1, p. 40, Lem. 4.1, p. 42). The short form suffices for construction. □

The algorithm for the determination of bicyclic bicubic fields has been presented in [8] (Alg. 7, pp. 24–26), but $B_5,\dots ,B_{10}$ should be defined as in Formula (23) (short form without $k_1,\dots ,k_4$).

Corollary 2.

The  capitulation targets, i.e., the unramified cyclic cubic relative extensions of $k_1$, $k_2$, $k_3$, and $k_4$, respectively, among the absolutely bicyclic bicubic subfields of the 3-genus field $k^{*}=k_p{k}_q{k}_r$ are $B_1,B_5,B_6,B_7$; $B_2,B_7,B_8,B_9$; $B_3,B_5,B_9,B_{10}$; and $B_4,B_6,B_8,B_{10}$, respectively.

In particular, $B_7$ is common to both $k_1$ and $k_2$, $B_5$ is common to $k_1$ and $k_3$, $B_6$ is common to $k_1$ and $k_4$, $B_9$ is common to $k_2$ and $k_3$, $B_8$ is common to $k_2$ and $k_4$, and $B_{10}$ is common to $k_3$ and $k_4$.

Proof. 

This follows immediately from Theorem 8, Formulas (22) and (23). □

Proposition 1.

If there exists $1\le j\le 10$ such that $h_3\left(B_j\right)=3$, then $h_3\left(B_{\mathsf{\ell }}\right)=3$, for all $1\le \mathsf{\ell }\le 10$, and $h_3\left(k_i\right)=9$, for all $1\le i\le 4$.

The 3-class number of $B_j$, $1\le j\le 10$, satisfies the  tame   condition $h_3\left(B_j\right)=(U_j:V_j)$ if and only if for each cyclic cubic subfield k of $B_j$ the Hilbert 3-class field ${\mathrm{F}}_3^1\left(k\right)$ of k coincides with the genus field $k^{*}$ of k. Otherwise, the  wild   condition $h_3\left(B_j\right)>(U_j:V_j)$ holds.

If there exists $1\le j\le 10$ such that $h_3\left(B_j\right)>(U_j:V_j)$, then $9\mid h_3\left(B_{\mathsf{\ell }}\right)$, for all $1\le \mathsf{\ell }\le 10$.

Proof. 

The condition is trivial for the subfields k with $t=1$, since $h_3\left(k\right)=[{\mathrm{F}}_3^1\left(k\right):k]=[k^{*}:k]=1$ is satisfied anyway. However, the subfields k with $t=2$ must have the 3-class number $h_3\left(k\right)=[{\mathrm{F}}_3^1\left(k\right):k]=[k^{*}:k]=3$, in particular, the prime divisors of the conductor are not mutual cubic residues, and the subfields k with $t=3$ must have the 3-class number $h_3\left(k\right)=[{\mathrm{F}}_3^1\left(k\right):k]=[k^{*}:k]=9$, that is, they cannot have the 3-class rank $\varrho \left(k\right)\ge 3$. For details, see [2] (pp. 47–48, i.p. Prop. 4.5). □

Let $t=3$ and $k_{\mu }$, $1\le \mu \le 4$ be one of the four cyclic cubic number fields sharing the common conductor $c=pqr$, and suppose $B_j$, $1\le j\le 10$, is one of the ten bicyclic bicubic subfields of the absolute 3-genus field $k^{*}$ of $k_{\mu }$ such that $B_j/k_{\mu }$ is an unramified cyclic extension of degree 3. We denote by $U_j$ the unit group of $B_j$, by $V_j$ the subgroup generated by all subfield units, by $r_j$ the rank of the principal factor matrix $M_j$ of $B_j$, and by $A=\left(a_{\iota \lambda }\right)$ the right upper triangular $(8\times 8)$-matrix such that $({\gamma }_1^3,\dots ,{\gamma }_8^3)=({\epsilon }_1,\dots ,{\epsilon }_8)·A$ (in the sense of exponentiation), for a suitable torsion-free basis $({\gamma }_1,\dots ,{\gamma }_8)$ of $U_j$ and a canonical basis $({\epsilon }_1,\dots ,{\epsilon }_8)$ of $V_j$, according to [2,4] (pp. 497–503) (pp. 19–22).

For several times, Ayadi [2] alludes to the following fact: the minimal subfield unit index $(U_j:V_j)=3$ for the matrix rank $r_j=3$ of $B_j$ corresponds to the maximal unit norm index $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3$, associated with a total transfer kernel $\#ker\left(T_{B_j/k_{\mu }}\right)=9$ of $B_j/k_{\mu }$. Since he does not give a proof, we summarize all related issues in a lemma.

Lemma 2.

The following statements are equivalent, row by row:

$$\begin{array}{cc}\hfill (U_j:V_j)=3& ⟺a_{77}=3,a_{88}=3,a_{66}=1⟹(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3,\hfill \\ \hfill (U_j:V_j)=9& ⟺a_{77}=3,a_{88}=1,a_{66}=1⟹(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3,\hfill \\ \hfill (U_j:V_j)=27& ⟺a_{77}=1,a_{88}=1,a_{66}=1⟺(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=1.\hfill \end{array}$$

(25)

Proof. 

According to Theorem 5, $r_j=3$⟺$(U_j:V_j)=3$, and $r_j=2$⟺$(U_j:V_j)\in \{9,27\}$.

• Now, $a_{77}=1$ implies ${\gamma }_7^3=\left({\prod }_{\iota =1}^6{\epsilon }_{\iota }^{a_{\iota 7}}\right)·{\epsilon }_7$, $N_{B_j/k_{\mu }}\left({\gamma }_7\right)=\pm {\epsilon }_7$, $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=1$,
• but $a_{77}=3$ implies ${\gamma }_7^3=\left({\prod }_{\iota =1}^6{\epsilon }_{\iota }^{a_{\iota 7}}\right)·{\epsilon }_7^3$, $N_{B_j/k_{\mu }}\left({\gamma }_7\right)=\pm {\epsilon }_7^3$, $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3$.

Finally, Theorem 7 on the Herbrand quotient of $U_j$ shows the cardinality of the transfer kernel, $\#ker{T}_{B_j/k_{\mu }}=[k_{\mu }:\mathbb{Q}]·(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3·(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))$. □

Proposition 2.

Let ℓ be an odd prime, and suppose that $B=K·L$ is a bicyclic field of degree ${\mathsf{\ell }}^2$, compositum of two cyclic fields K and L of degree ℓ. If p is a prime number that ramifies in both, K and L, i.e., $p{\mathcal{O}}_K={\mathfrak{p}}_1^{\mathsf{\ell }}$ and $p{\mathcal{O}}_L={\mathfrak{p}}_2^{\mathsf{\ell }}$, then the extension ideals ${\mathfrak{p}}_1{\mathcal{O}}_B={\mathfrak{p}}_2{\mathcal{O}}_B$ coincide.

Proof. 

If the decomposition invariants of p in B are $(e,f,g)=(\mathsf{\ell },1,\mathsf{\ell })$, resp. $(\mathsf{\ell },\mathsf{\ell },1)$, resp. $({\mathsf{\ell }}^2,1,1)$, then those of ${\mathfrak{p}}_1$ and ${\mathfrak{p}}_2$ in B must be identical $(e,f,g)=(1,1,\mathsf{\ell })$, resp. $(1,\mathsf{\ell },1)$, resp. $(\mathsf{\ell },1,1)$, and unique prime decomposition enforces ${\mathfrak{p}}_1{\mathcal{O}}_B={\mathfrak{p}}_2{\mathcal{O}}_B$. □

Corollary 3.

Let $\mu \in \{1,2,3,4\}$ and $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$. Then, the following  capitulation laws   for ideal classes hold independently of the combined cubic residue symbol ${[p,q,r]}_3$.

1.

$\left[\mathfrak{p}\right]$ capitulates in $B_5/k_{\mu }$, for $\mu =1,3$, and in $B_8/k_{\mu }$, for $\mu =2,4$.

2.

$\left[\mathfrak{q}\right]$ capitulates in $B_6/k_{\mu }$, for $\mu =1,4$, and in $B_9/k_{\mu }$, for $\mu =2,3$.

3.

$\left[\mathfrak{r}\right]$ capitulates in $B_7/k_{\mu }$, for $\mu =1,2$, and in $B_{10}/k_{\mu }$, for $\mu =3,4$.

Proof. 

We show that $\left[\mathfrak{p}\right]\in {\mathrm{Cl}}_3\left(k_1\right)$ capitulates in $B_5$. Everything else is proved in the same way, always using Proposition 2 with $\mathsf{\ell }=3$. The bicyclic bicubic field $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$ is a compositum of the cyclic cubic fields $k_p$ and $k_1$. Since the conductor of $k_p$ is p, the principal factor $A\left(k_p\right)=p$ is determined uniquely, and $p{\mathcal{O}}_{k_p}={\mathfrak{p}}_0^3$ is totally ramified, whence ${\mathfrak{p}}_0=\alpha {\mathcal{O}}_{k_p}$ with $\alpha \in k_p^{\times }$ is necessarily a principal ideal. Since the conductor of $k_1$ is $c=pqr$, the prime $p{\mathcal{O}}_{k_1}={\mathfrak{p}}^3$ is also totally ramified, and Proposition 2 asserts that $\mathfrak{p}{\mathcal{O}}_{B_5}={\mathfrak{p}}_0{\mathcal{O}}_{B_5}$, which is the principal ideal $\alpha {\mathcal{O}}_{B_5}$. Thus, the class $\left[\mathfrak{p}\right]$ capitulates in $B_5$. □

Proposition 3.

If ${\left(\frac{p}{q}\right)}_3=1$ but ${\left(\frac{q}{p}\right)}_3\ne 1$, then ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq \left(3\right)$, ${\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)\simeq \left(3\right)$, and  two principal factors   are given by $A\left(k_{pq}\right)=p$, $A\left({\tilde{k}}_{pq}\right)=p$.

Proof. 

If $p\to q$, then p splits in $k_q$, $p{\mathcal{O}}_{k_q}={\wp }_1{\wp }_2{\wp }_3$, and ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq \left(3\right)$, according to Georges Gras [1]. The Hilbert 3-class field ${\mathrm{F}}_3^1\left(k_{pq}\right)$ of $k_{pq}$ with $[{\mathrm{F}}_3^1\left(k_{pq}\right):k_{pq}]=3$ coincides with the absolute 3-genus field $k^{*}=k_p·k_q=k_{pq}·{\tilde{k}}_{pq}$ of the doublet $(k_{pq},{\tilde{k}}_{pq})$ with $[k^{*}:\mathbb{Q}]=9$ and $[k^{*}:k_{pq}]=3$.

Since the conductor of $k_{pq}$ is $pq$, $p{\mathcal{O}}_{k_{pq}}={\mathfrak{p}}^3$ is ramified in $k_{pq}$, but $k^{*}$ is unramified over $k_{pq}$, and the decomposition invariants of p in $k^{*}$ are $(e,f,g)=(3,1,3)$, those of $\mathfrak{p}$ in $k^{*}={\mathrm{F}}_3^1\left(k_{pq}\right)$ are $(e,f,g)=(1,1,3)$, i.e., $\mathfrak{p}$ splits completely in ${\mathrm{F}}_3^1\left(k_{pq}\right)$,

By the decomposition law of the Hilbert 3-class field, $\mathfrak{p}=\alpha {\mathcal{O}}_{k_{pq}}$ is principal with $\alpha \in k_{pq}^{\times }$. Therefore, the unique principal factor of $k_{pq}$ is $A\left(k_{pq}\right)=p$. The same reasoning is true for ${\tilde{k}}_{pq}$. □

Proposition 4.

Let $\mu \in \{1,2,3,4\}$, such that ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$.

• If ${\left(\frac{p}{q}\right)}_3=1$ and ${\left(\frac{p}{r}\right)}_3=1$, then  the principal factor   of $k_{\mu }$ is $A\left(k_{\mu }\right)=p$.

Proof. 

Since ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$, the Hilbert 3-class field ${\mathrm{F}}_3^1\left(k_{\mu }\right)$ of $k_{\mu }$ with $[{\mathrm{F}}_3^1\left(k_{\mu }\right):k_{\mu }]=9$ coincides with the absolute 3-genus field $k^{*}=k_p·k_q·k_r$ of the quartet $(k_1,\dots ,k_4)$ with $[k^{*}:\mathbb{Q}]=27$ and $[k^{*}:k_{\mu }]=9$.

Since the conductor of $k_{\mu }$ is $c=pqr$, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$ is ramified in $k_{\mu }$, but $k^{*}$ is unramified over $k_{\mu }$.

If $q\leftarrow p\to r$ is universally repelling, then p splits in $k_q$ and in $k_r$, and the decomposition invariants of p in $k^{*}$ are $(e,f,g)=(3,1,9)$; those of $\mathfrak{p}$ in $k^{*}={\mathrm{F}}_3^1\left(k_{\mu }\right)$ are $(e,f,g)=(1,1,9)$, i.e., $\mathfrak{p}$ splits completely in ${\mathrm{F}}_3^1\left(k_{\mu }\right)$; and the decomposition law of the Hilbert 3-class field implies that $\mathfrak{p}=\alpha {\mathcal{O}}_{k_{\mu }}$ is principal with $\alpha \in k_{\mu }^{\times }$. Therefore, the unique principal factor of $k_{\mu }$ is $A\left(k_{\mu }\right)=p$. □

5. Finite 3-Groups of Type (3,3)

In the following tables, we list those invariants of finite 3-groups G with elementary bicyclic commutator quotient $G/G^{\prime }\simeq (3,3)$ which qualify metabelian groups $\mathfrak{M}$ as second 3-class groups $\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ and non-metabelian groups $\mathfrak{G}$ as 3-class field tower groups $\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k\right)/k)$ of cyclic cubic number fields k. The process of searching for suitable groups in descendant trees with the strategy of pattern recognition [15] is governed by the Artin pattern $\mathrm{AP}=(\alpha ,\varkappa )$ [16] (p. 27), where $\alpha ={\alpha }_1$ and $\varkappa ={\varkappa }_1$, respectively, denote the first layer of the transfer target type (TTT) and the transfer kernel type (TKT), respectively. Additionally, we give the top layer ${\alpha }_2$ of the TTT, which consists of the abelian quotient invariants of the commutator subgroup ${\mathfrak{M}}^{\prime }$, corresponding to the 3-class group of the first Hilbert 3-class field ${\mathrm{F}}_3^1\left(k\right)$ of k. The nuclear rank $\nu$ is responsible for the search complexity. The p-multiplicator rank $\mu$ of a group G is precisely its relation rank $d_2\left(G\right)={dim}_{{\mathbb{F}}_3}{\mathrm{H}}^2(G,{\mathbb{F}}_3)$, which decides whether G is admissible as $\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k\right)/k)$, according to the Shafarevich Theorem [16,17]. In the case of cyclic cubic fields k, it is limited by the Shafarevich bound $\mu \le \varrho +r+\theta$, where $\varrho =d_1\left(G\right)={dim}_{{\mathbb{F}}_3}{\mathrm{H}}^1(G,{\mathbb{F}}_3)$ denotes the generator rank of G, which coincides with the 3-class rank $\varrho$ of k, $r=r_1+r_2-1=2$ is the torsion-free Dirichlet unit rank of the field k with signature $(r_1,r_2)=(3,0)$, and $\theta =0$ indicates the absence of a (complex) primitive third root of unity in the totally real field k. Finally, $\pi \left(\mathfrak{M}\right)=\mathfrak{M}/{\gamma }_c\left(\mathfrak{M}\right)$ denotes the parent of $\mathfrak{M}$, that is the quotient by the last non-trivial lower central with $c=\mathrm{cl}\left(\mathfrak{M}\right)$.

Theorem 9.

Let k be a cyclic cubic number field with elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k\right)\simeq (3,3)$. Denote by $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ the second 3-class group of k, and by $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k\right)/k)$ the 3-class field tower group of k. Then, the Artin pattern $(\alpha ,\varkappa )$ of k identifies the groups $\mathfrak{M}$ and $\mathfrak{G}$, and determines the length ${\mathsf{\ell }}_3\left(k\right)$ of the 3-class field tower of k, according to the following deterministic laws. (See the associated descendant tree ${\mathcal{T}}^1\langle 9,2\rangle$ in [8] (Fig. 6.1, p. 44).)

1.

If $\alpha =[1,1,1,1]$, $\varkappa =\left(0000\right)$ (type $\mathrm{a}.1$), then $\mathfrak{G}\simeq \langle 9,2\rangle$ and ${\mathsf{\ell }}_3\left(k\right)=1$.

2.

If $\alpha \sim [11,2,2,2]$, $\varkappa \sim \left(1111\right)$ (type $\mathrm{A}.1$), then $\mathfrak{G}\simeq \langle 27,4\rangle$.

3.

If $\alpha \sim [111,11,11,11]$, $\varkappa \sim \left(2000\right)$ (type $\mathrm{a}.3^{*}$), then $\mathfrak{G}\simeq \langle 81,7\rangle$.

4.

If $\alpha \sim [21,11,11,11]$, $\varkappa \sim \left(2000\right)$ (type $\mathrm{a}.3$), then $\mathfrak{G}\simeq \langle 81,8\rangle$.

5.

If $\alpha \sim [21,11,11,11]$, $\varkappa \sim \left(1000\right)$ (type $\mathrm{a}.2$), then $\mathfrak{G}\simeq \langle 81,10\rangle$.

6.

If $\alpha \sim [22,11,11,11]$, $\varkappa \sim \left(2000\right)$ (type $\mathrm{a}.3$), then $\mathfrak{G}\simeq \langle 243,25\rangle$.

7.

If $\alpha \sim [22,11,11,11]$, $\varkappa \sim \left(1000\right)$ (type $\mathrm{a}.2$), then $\mathfrak{G}\simeq \langle 243,27\rangle$.

Except for the abelian tower in item (1), the tower is metabelian with ${\mathsf{\ell }}_3\left(k\right)=2$.

Proof. 

Generally, a cyclic cubic field k has the signature $(r_1,r_2)=(3,0)$ and the torsion-free unit rank $r=r_1+r_2-1=2$, does not contain primitive third roots of unity, and thus possesses the maximal admissible relation rank $d_2\le d_1+r=4$ for the group $\mathfrak{G}$, when its 3-class rank, i.e., the generator rank of $\mathfrak{G}$, is $d_1=\varrho =2$. Consequently, ${\mathsf{\ell }}_3\left(k\right)\ge 3$ in the case of $d_2\left(\mathfrak{M}\right)\ge 5$.

For item (1), we have $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)\simeq \langle 9,2\rangle \simeq (3,3)\simeq {\mathrm{Cl}}_3\left(k\right)\simeq \mathrm{Gal}({\mathrm{F}}_3^1\left(k\right)/k)$, whence ${\mathsf{\ell }}_3\left(k\right)=1$. We always identify groups according to [3,18].

For item (2) to item (7), the group $\mathfrak{M}$ is of maximal class (coclass $\mathrm{cc}\left(\mathfrak{M}\right)=1$), and thus coincides with $\mathfrak{G}$, whence ${\mathsf{\ell }}_3\left(k\right)=2$.

In each case, the Artin pattern $(\alpha ,\varkappa )$ identifies $\mathfrak{M}=\mathfrak{G}$ uniquely, and the relation ranks are $d_2\langle 9,2\rangle =3$, $d_2\langle 27,4\rangle =2$, $d_2\langle 81,7\rangle =3$, $d_2\langle 81,8\rangle =3$, $d_2\langle 81,10\rangle =3$, $d_2\langle 243,25\rangle =3$, $d_2\langle 243,27\rangle =3$, each of them less than 4. □

Corollary 4.

Under the assumptions of Theorem 9, the abelian-type invariants ${\alpha }_2$ of the 3-class group ${\mathrm{Cl}}_3\left({\mathrm{F}}_3^1\left(k\right)\right)$ of the first Hilbert 3-class field of k are required for the unambiguous identification of the following groups: $\mathfrak{G}$ and $\mathfrak{M}$, respectively. (See the associated descendant tree ${\mathcal{T}}^2\langle 729,40\rangle$ in [8] (Fig. 6.2, p. 45).)

• If $\alpha \sim [21,11,11,11]$, $\varkappa =\left(0000\right)$, $\mathrm{a}.1$, then $\mathfrak{G}\simeq \left\{\begin{array}{cc}\langle 81,9\rangle \hfill & \mathit{for}{\alpha }_2=\left[11\right],\hfill \\ \langle 243,28..30\rangle \hfill & \mathit{for}{\alpha }_2\sim \left[21\right].\hfill \end{array}\right.$
• If $\alpha \sim [21,21,111,111]$, $\varkappa \sim \left(0043\right)$, $\mathrm{b}.10$, then $\mathfrak{M}\simeq \left\{\begin{array}{cc}\langle 729,34..36\rangle \hfill & \mathit{for}{\alpha }_2=\left[1111\right],\hfill \\ \langle 729,37..39\rangle \hfill & \mathit{for}{\alpha }_2\sim \left[211\right].\hfill \end{array}\right.$

Proof. 

The Artin pattern $(\alpha ,\varkappa )$ of k alone is not able to identify the groups $\mathfrak{M}$ and $\mathfrak{G}$ unambiguously. Ascione [19] uses the notation $\langle 729,34\rangle =H$, $\langle 729,35\rangle =I$, $\langle 729,37\rangle =A$, $\langle 729,38\rangle =C$. □

In

Table 2. Invariants of metabelian 3-groups $\mathfrak{M}$ with $\mathfrak{M}/{\mathfrak{M}}^{\prime }\simeq (3,3)$.

$\mathfrak{M}$	cc	Type	$\mathit{\varkappa }$	$\mathit{\alpha }$	${\mathit{\alpha }}_2$	$\mathit{\nu }$	$\mathit{\mu }$	$\mathit{\pi }\left(\mathfrak{M}\right)$
$\langle 9,2\rangle$	1	a.1	$\left(0000\right)$	$1,1,1,1$	0	3	3	—
$\langle 27,3\rangle$	1	a.1	$\left(0000\right)$	$11,11,11,11$	1	2	4	$\langle 9,2\rangle$
$\langle 27,4\rangle$	1	A.1	$\left(1111\right)$	$11,2,2,2$	1	0	2	$\langle 9,2\rangle$
$\langle 81,7\rangle$	1	a.3${}^{*}$	$\left(2000\right)$	$111,11,11,11$	11	0	3	$\langle 27,3\rangle$
$\langle 81,8\rangle$	1	a.3	$\left(2000\right)$	$21,11,11,11$	11	0	3	$\langle 27,3\rangle$
$\langle 81,9\rangle$	1	a.1	$\left(0000\right)$	$21,11,11,11$	11	1	4	$\langle 27,3\rangle$
$\langle 81,10\rangle$	1	a.2	$\left(1000\right)$	$21,11,11,11$	11	0	3	$\langle 27,3\rangle$
$\langle 243,25\rangle$	1	a.3	$\left(2000\right)$	$22,11,11,11$	21	0	3	$\langle 81,9\rangle$
$\langle 243,27\rangle$	1	a.2	$\left(1000\right)$	$22,11,11,11$	21	0	3	$\langle 81,9\rangle$
$\langle 243,28..30\rangle$	1	a.1	$\left(0000\right)$	$21,11,11,11$	21	0	3	$\langle 81,9\rangle$
$\langle 243,3\rangle$	2	b.10	$\left(0043\right)$	$21,21,111,111$	111	2	4	$\langle 27,3\rangle$
$\langle 729,34\rangle =H$	2	b.10	$\left(0043\right)$	$21,21,111,111$	1111	2	5	$\langle 243,3\rangle$
$\langle 729,35\rangle =I$	2	b.10	$\left(0043\right)$	$21,21,111,111$	1111	1	4	$\langle 243,3\rangle$
$\langle 729,37\rangle =A$	2	b.10	$\left(0043\right)$	$21,21,111,111$	211	2	5	$\langle 243,3\rangle$
$\langle 729,38\rangle =C$	2	b.10	$\left(0043\right)$	$21,21,111,111$	211	1	4	$\langle 243,3\rangle$
$\langle 729,40\rangle =B$	2	b.10	$\left(0043\right)$	$22,21,111,111$	211	2	5	$\langle 243,3\rangle$
$\langle 729,41\rangle =D$	2	d.19	$\left(4043\right)$	$22,21,111,111$	211	1	4	$\langle 243,3\rangle$
$\langle 729,42\rangle$	2	d.23	$\left(1043\right)$	$22,21,111,111$	211	0	3	$\langle 243,3\rangle$
$\langle 729,43\rangle$	2	d.25	$\left(2043\right)$	$22,21,111,111$	211	0	3	$\langle 243,3\rangle$
$\langle 2187,248|249\rangle$	2	d.19	$\left(4043\right)$	$32,21,111,111$	221	0	4	$\langle 729,40\rangle$
$\langle 2187,250\rangle$	2	d.23	$\left(1043\right)$	$32,21,111,111$	221	0	4	$\langle 729,40\rangle$
$\langle 2187,251|252\rangle$	2	d.25	$\left(2043\right)$	$32,21,111,111$	221	0	4	$\langle 729,40\rangle$
$\langle 2187,253\rangle$	2	b.10	$\left(0043\right)$	$22,21,111,111$	221	1	5	$\langle 729,40\rangle$
$\langle 6561,1989\rangle$	2	d.19	$\left(4043\right)$	$33,21,111,111$	321	0	4	$\langle 2187,247\rangle$
$\langle 243,8\rangle$	2	c.21	$\left(0231\right)$	$21,21,21,21$	111	1	3	$\langle 27,3\rangle$
$\langle 729,52\rangle =S$	2	G.16	$\left(4231\right)$	$22,21,21,21$	211	1	3	$\langle 243,8\rangle$
$\langle 729,54\rangle =U$	2	c.21	$\left(0231\right)$	$22,21,21,21$	211	2	4	$\langle 243,8\rangle$
$\langle 2187,301|305\rangle$	2	G.16	$\left(4231\right)$	$32,21,21,21$	221	1	4	$\langle 729,54\rangle$
$\langle 2187,303\rangle$	2	c.21	$\left(0231\right)$	$32,21,21,21$	221	1	4	$\langle 729,54\rangle$
$\langle 2187,64\rangle =P_7$	3	b.10	$\left(0043\right)$	$22,22,111,111$	2111	4	6	$\langle 243,3\rangle$
$\langle 2187,65|67\rangle$	3	H.4	$\left(3343\right)$	$22,22,111,111$	2111	3	5	$\langle 243,3\rangle$
$\langle 2187,66|73\rangle$	3	F.11	$\left(1143\right)$	$22,22,111,111$	2111	2	4	$\langle 243,3\rangle$
$\langle 2187,69\rangle$	3	G.16	$\left(1243\right)$	$22,22,111,111$	2111	2	4	$\langle 243,3\rangle$
$\langle 2187,71\rangle$	3	G.19	$\left(2143\right)$	$22,22,111,111$	2111	2	4	$\langle 243,3\rangle$
$\langle 6561,676|677\rangle$	3	d.19	$\left(4043\right)$	$32,22,111,111$	2211	0	5	$\langle 2187,64\rangle$
$\langle 6561,678\rangle$	3	d.23	$\left(1043\right)$	$32,22,111,111$	2211	0	5	$\langle 2187,64\rangle$
$\langle 6561,679|680\rangle$	3	d.25	$\left(2043\right)$	$32,22,111,111$	2211	0	5	$\langle 2187,64\rangle$
$\langle 6561,693..698\rangle$	3	b.10	$\left(0043\right)$	$22,22,111,111$	2211	0	5	$\langle 2187,64\rangle$
$P_7-\#2;34|35$	4	H.4	$\left(3343\right)$	$32,32,111,111$	2221	1	5	$\langle 2187,64\rangle$

, we begin with metabelian groups $\mathfrak{M}$ of generator rank $d_1\left(\mathfrak{M}\right)=2$. The Shafarevich bound [16] (Thm. 5.1, p. 28) is given by $\mu \le \varrho +r+\theta =2+2+0=4$. For order 6561 see [20].

Capital letters for $\mathfrak{M}$ are due to Ascione [19]. For the metabelian groups $\mathfrak{M}$ with non-trivial cover $\mathrm{cov}\left(\mathfrak{M}\right)$ [16] (p. 30), we need non-metabelian groups $\mathfrak{G}$ in the cover, which are given in

Table 3. Invariants of non-metabelian 3-groups $\mathfrak{G}$ with $\mathfrak{G}/{\mathfrak{G}}^{\prime }\simeq (3,3)$.

$\mathfrak{G}$	cc	Type	$\mathit{\varkappa }$	$\mathit{\alpha }$	${\mathit{\alpha }}_2$	$\mathit{\nu }$	$\mathit{\mu }$	$\mathfrak{G}/{\mathfrak{G}}^{″}$
$\langle 2187,263..265\rangle$	2	d.19	$\left(4043\right)$	$22,21,111,111$	211	0	3	$\langle 729,41\rangle$
$\langle 2187,307|308\rangle$	2	c.21	$\left(0231\right)$	$22,21,21,21$	211	0	3	$\langle 729,54\rangle$
$\langle 6561,619|623\rangle$	3	G.16	$\left(4231\right)$	$32,21,21,21$	221	1	3	$\langle 2187,$301|305〉

, where we begin with groups $\mathfrak{G}$ of generator rank $d_1\left(\mathfrak{G}\right)=2$. For $d_1\left(\mathfrak{G}\right)=3$, we refer to a forthcoming paper. Instead of the parent $\pi \left(\mathfrak{G}\right)$, we give the metabelianization $\mathfrak{G}/{\mathfrak{G}}^{″}$.

6. Categories I and II

A common feature of these two categories is the inhomogeneity of 3-class ranks of the four components in the quartet ${\left(k_{\mu }\right)}_{\mu =1}^4$ sharing the conductor $c=pqr$. In the present article, we restrict ourselves to 3 and 2 components, respectively, with elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$, for Category $\mathrm{I}$ and Category $\mathrm{II}$, respectively, and we postpone elementary tricyclic ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3,3)$ to a future paper. All computations for examples were performed with Magma [6,21,22].

Definition 5.

According to the 3-class numbers $h_3\left(k_{\mu }\right)$, a quartet ${\left(k_{\mu }\right)}_{\mu =1}^4$ of cyclic cubic fields with common conductor $c=pqr$ belonging to Category $\mathrm{I}$ or $\mathrm{II}$ is called

$$\left\{\begin{array}{cc}\mathit{regular}\hfill & \\ \mathit{singular}\hfill & \\ \mathit{super}-\mathit{singular}\hfill & \end{array}\right.\mathit{if}max\{h_3\left(k_{\mu }\right)\mid 1\le \mu \le 4\}\left\{\begin{array}{cc}=27,\hfill & \\ =81,\hfill & \\ \ge 243.\hfill & \end{array}\right.$$

(26)

In a regular, singular, and super-singular quartet, respectively, there occur 3-class groups ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3,3)$, ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (9,3,3)$, and ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (9,9,3)$, respectively, for some $1\le \mu \le 4$.

6.1. Category I, Graph 1

Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 1 of Category $\mathrm{I}$ with combined cubic residue symbol ${[p,q,r]}_3=\{p,q,r;\delta \equiv 0\left(\mathrm{mod}3\right)\}$.

Since there are no trivial cubic residue symbols among the three prime(power) divisors $p,q,r$ of the conductor $c=pqr$, the principal factors of the subfields with $t=2$ of the absolute genus field $k^{*}$ must be divisible by both relevant primes [2] (Prop. 3.2, p. 26), and we can use the general approach

$$\begin{array}{cc}\hfill A\left(k_{pq}\right)=p^{\mathsf{\ell }}q,& A\left({\tilde{k}}_{pq}\right)=p^{-\mathsf{\ell }}q,\hfill \\ \hfill A\left(k_{pr}\right)=p^{m}r,& A\left({\tilde{k}}_{pr}\right)=p^{-m}r,\mathrm{and}\hfill \\ \hfill A\left(k_{qr}\right)=q^{n}r,& A\left({\tilde{k}}_{qr}\right)=q^{-n}r,\hfill \end{array}$$

(27)

with $\mathsf{\ell },m,n\in \{-1,1\}$, identifying $-1\equiv 2\left(\mathrm{mod}3\right)$, since it is easier to manage: ${\mathsf{\ell }}^2=m^2=n^2=1$.

Lemma 3.

The product $\mathsf{\ell }·m·n=-1$ is negative (that is, either one or three among $\mathsf{\ell },m,n$ are negative) if and only if the compositum $L=k_{pq}{k}_{pr}{k}_{qr}$ satisfies the normalization $[L:\mathbb{Q}]=9$:

$$\begin{array}{cc}\hfill \mathsf{\ell }·m·n=-1& ⟺[L:\mathbb{Q}]=9,\hfill \\ \hfill \mathsf{\ell }·m·n=+1& ⟺[L:\mathbb{Q}]=27.\hfill \end{array}$$

(28)

Proof. 

By Theorem 8, the fields L and $\tilde{L}={\tilde{k}}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$ satisfy a skew balance of their degrees $\in \{9,27\}$ in the product $[L:\mathbb{Q}]·[\tilde{L}:\mathbb{Q}]=243$.

Suppose $lmn=+1$. Then, we produce a contradiction by the assumption that $[L:\mathbb{Q}]=9$ and $[\tilde{L}:\mathbb{Q}]=27$. We define the compositum $K:={\tilde{k}}_{pq}{\tilde{k}}_{pr}$ of degree 9. Then, K contains one of the fields $k_{\mu }$, $\mu =1,\dots ,4$, and either ${\tilde{k}}_{qr}$ or $k_{qr}$. In the former case, $K=\tilde{L}$ would have degree 27. So, $K={\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, and we calculate the following sub-determinants of the principal factor matrices $M_L$ and $M_K$, with respect to the fields with $t=2$ only (ignoring the field with $t=3$):

$\left|\begin{array}{ccc}\mathsf{\ell }& 1& 0\\ m& 0& 1\\ 0& n& 1\end{array}\right|=-\mathsf{\ell }n-m=0$$⟺$$\left|\begin{array}{ccc}-\mathsf{\ell }& 1& 0\\ -m& 0& 1\\ 0& n& 1\end{array}\right|=\mathsf{\ell }n+m=0$$⟺$$\mathsf{\ell }n=-m$.

However, $lmn=+1$ implies $\mathsf{\ell }n=m$, and thus rank 3 of $M_L$ and $M_K$. By (17), this gives indices of subfield units $(U_L:V_L)=3$ and $(U_K:V_K)=3$. At least one among L and K, say X, does not contain the critical field $k_{\mu }$ with ${\varrho }_3\left(k_{\mu }\right)=3$, whence it is tame with $h_3\left(X\right)=(U_X:V_X)=3$, in contradiction to $9\mid h_3\left(X\right)$, by Proposition 1. Thus, we must have $[L:\mathbb{Q}]=27$.

With nearly identical arguments, it is easy to show that $lmn=-1$ implies $[L:\mathbb{Q}]=9$. □

Lemma 4.

(3-class ranks of components for $\mathrm{I}.1$.)  Without loss of generality, precisely three components, $k_2$, $k_3$, $k_4$, of the quartet have elementary bicyclic 3-class groups ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$, $2\le \mu \le 4$, whereas the single remaining component, $k_1$, has the 3-class rank ${\varrho }_3\left(k_1\right)=3$. In dependence on the  decisive principal factors   in Formula (27), the principal factors of $k_{\mu }$ are

$$\begin{array}{cc}\hfill A\left(k_2\right)=p{q}^{2}r,A\left(k_3\right)=pqr,A\left(k_4\right)=pq{r}^2& \mathit{if}(\mathsf{\ell },m,n)=(1,1,2),\hfill \\ \hfill A\left(k_2\right)=p^{2}qr,A\left(k_3\right)=pq{r}^2,A\left(k_4\right)=pqr& \mathit{if}(\mathsf{\ell },m,n)=(1,2,1),\hfill \\ \hfill A\left(k_2\right)=pqr,A\left(k_3\right)=p{q}^{2}r,A\left(k_4\right)=p^{2}qr& \mathit{if}(\mathsf{\ell },m,n)=(2,1,1),\hfill \\ \hfill A\left(k_2\right)=pq{r}^2,A\left(k_3\right)=p^{2}qr,A\left(k_4\right)=p{q}^{2}r& \mathit{if}(\mathsf{\ell },m,n)=(2,2,2).\hfill \end{array}$$

(29)

The  tame   condition $9\mid h_3\left(B_j\right)=(U_j:V_j)\in \{9,27\}$ with $r_j=2$ is satisfied for $j\in \{2,3,4,8,9,10\}$.

A further  decisive principal factor $A\left(k_1\right)=p^{e_1}{q}^{e_2}{r}^{e_3}$ and the associated invariant counter $\mathcal{D}:=\#\{1\le i\le 3\mid e_i\ne 0\}$ admit several conclusions for  wild   ranks:

$$r_5=r_6=r_7=3\mathit{iff}\mathcal{D}=2\mathit{iff}A\left(k_1\right)\mathit{has}\mathit{precisely}\mathit{two}\mathit{prime}\mathit{divisors}.$$

(30)

Proof. 

According to [2] (Prop. 4.4, pp. 43–44), the required condition to distinguish the unique component $k_1$ with the 3-class rank $\varrho \left(k_1\right)=3$ in the quartet ${\left(k_{\mu }\right)}_{\mu =1}^4$ is the set of decomposition invariants $(e,f,g)=(3,1,3)$ simultaneously for $p,q,r$ in the bicyclic bicubic field $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, that is,

• p splits in $k_{qr}$, and thus also in $B_j$ for $j\in \{1,3,8\}$;
• q splits in $k_{pr}$, and thus also in $B_j$ for $j\in \{1,4,9\}$;
• r splits in $k_{pq}$, and thus also in $B_j$ for $j\in \{1,2,10\}$.

Then, exactly the six fields $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$, $B_{10}=k_3{k}_4{k}_r{k}_{pq}$ do not contain $k_1$, and satisfy the tame relation $9\mid h_3\left(B_j\right)=(U_j:V_j)\in \{9,27\}$ with ranks $r_j=2$ for $j=2,3,4,8,9,10$, by Proposition 1.

This fact can be exploited for each tame bicyclic bicubic field $B_j$, by calculating the rank $r_j$ with row operations on the associated principal factor matrix $M_j$ and drawing conclusions for the exponents $x_{\mu },y_{\mu },z_{\mu }$ in the approach $A\left(k_{\mu }\right)=p^{x_{\mu }}{q}^{y_{\mu }}{r}^{z_{\mu }}$, $1\le \mu \le 4$:

$M_8=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ x_4& y_4& z_4\\ 1& 0& 0\\ 0& n& 1\end{array}\right)$, $M_9=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ x_3& y_3& z_3\\ 0& 1& 0\\ m& 0& 1\end{array}\right)$, $M_{10}=\left(\begin{array}{ccc}{x}_3& y_3& z_3\\ x_4& y_4& z_4\\ 0& 0& 1\\ \mathsf{\ell }& 1& 0\end{array}\right)$.

For $B_8=k_2{k}_4{k}_p{k}_{qr}$, $M_8$ leads to decisive pivot elements $z_2-n{y}_2$ and $z_4-n{y}_4$ in the last column; for $B_9=k_2{k}_3{k}_q{k}_{pr}$, $M_9$ leads to $z_2-m{x}_2$ and $z_3-m{x}_3$ in the last column; and for $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $M_{10}$ leads to $y_3-\mathsf{\ell }{x}_3$ and $y_4-\mathsf{\ell }{x}_4$ in the middle column.

So, $r_8=r_9=r_{10}=2$ implies $n{y}_2\equiv z_2$, $n{y}_4\equiv z_4$, $z_2\equiv m{x}_2$, $z_3\equiv m{x}_3$, $\mathsf{\ell }{x}_3\equiv y_3$, $\mathsf{\ell }{x}_4\equiv y_4$. Or, in combined form, $m{x}_2\equiv n{y}_2\equiv z_2$, $m{x}_3\equiv -n{y}_3\equiv z_3$, $-m{x}_4\equiv n{y}_4\equiv z_4$. This yields (29).

Additionally, we use the remaining three tame ranks for

$M_2=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ \mathsf{\ell }& 1& 0\\ -m& 0& 1\\ 0& -n& 1\end{array}\right)$, $M_3=\left(\begin{array}{ccc}{x}_3& y_3& z_3\\ -\mathsf{\ell }& 1& 0\\ -m& 0& 1\\ 0& n& 1\end{array}\right)$, $M_4=\left(\begin{array}{ccc}{x}_4& y_4& z_4\\ -\mathsf{\ell }& 1& 0\\ m& 0& 1\\ 0& -n& 1\end{array}\right)$.

For $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $M_2$ leads to the decisive pivot elements $z_2+m{x}_2+n{y}_2$, $\mathsf{\ell }m+n$ in the last column; for $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $M_3$ leads to $z_3+m{x}_3-n{y}_3$, $-\mathsf{\ell }m-n$ in the last column; and for $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_4$ leads to $z_4-m{x}_4+n{y}_4$, $\mathsf{\ell }m+n$ in the last column. So, $r_2=r_3=r_4=2$ implies $m{x}_2+n{y}_2\equiv -z_2$, $m{x}_3-n{y}_3\equiv -z_3$, $-m{x}_4+n{y}_4\equiv -z_4$, since the other pivot elements vanish a priori, $\mathsf{\ell }m+n=0$, i.e., $\mathsf{\ell }m=-n$, because $\mathsf{\ell }mn=-1$ and $n^2=1$ in Lemma 3. The congruences follow already from those for $r_8=r_9=r_{10}=2$.

For each wild bicyclic bicubic field $B_j$, $j\in \{1,5,6,7\}$, the rank $r_j$ is now calculated with row operations on the associated principal factor matrix $M_j$:

$M_1=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ \mathsf{\ell }& 1& 0\\ m& 0& 1\\ 0& n& 1\end{array}\right)$, $M_5=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_3& y_3& z_3\\ 1& 0& 0\\ 0& -n& 1\end{array}\right)$, $M_6=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_4& y_4& z_4\\ 0& 1& 0\\ -m& 0& 1\end{array}\right)$, $M_7=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_2& y_2& z_2\\ 0& 0& 1\\ -\mathsf{\ell }& 1& 0\end{array}\right)$.

For $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $M_1$ leads to the decisive pivot element $z_1-m{x}_1-n{y}_1$, since $-\mathsf{\ell }m-n\equiv 0$. So, $r_1=2$ implies $z_1\equiv m{x}_1+n{y}_1$. For $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $M_5$ leads to $z_1+n{y}_1$, $z_3+n{y}_3$ in the last column. So, $r_5=3$ iff either $-z_1\not\equiv n{y}_1$ or $-z_3\not\equiv n{y}_3$ modulo 3. For $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $M_6$ leads to $z_1+m{x}_1$, $z_4+m{x}_4$. So, $r_6=3$ iff either $-z_1\not\equiv m{x}_1$ or $-z_4\not\equiv m{x}_4$. For $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ leads to $y_1+\mathsf{\ell }{x}_1$, $y_2+\mathsf{\ell }{x}_2$ in the middle column. So, $r_7=3$ iff either $-y_1\not\equiv \mathsf{\ell }{x}_1$ or $-y_2\not\equiv \mathsf{\ell }{x}_2$. For each of these three ranks, the second condition can never be satisfied.

Since at most one of the exponents $x_1,y_1,z_1$ may vanish, the new congruences immediately lead to (30). For instance, $z_1=0$⇒ $-z_1=0\not\equiv n{y}_1$, $-z_1=0\not\equiv m{x}_1$⇒ $r_5=r_6=3$; but $0=z_1\equiv m{x}_1+n{y}_1$ also implies $m{x}_1\equiv -n{y}_1$, $mn{x}_1\equiv -y_1$, $\mathsf{\ell }{x}_1\equiv y_1$, and thus $r_7=3$. Conversely, suppose $\mathcal{D}=3$. If $-z_1\not\equiv n{y}_1$, then $z_1\equiv n{y}_1$, and $z_1\equiv m{x}_1+n{y}_1$ implies $m{x}_1\equiv 0$, and thus the contradiction $x_1=0$. □

Proposition 5

(Sub-triplet with 3-rank two for $\mathrm{I}.1$). For fixed $\mu \in \{2,3,4\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$; then, the 3-class group of $k_{\mu }$ is generated by any two among the non-trivial classes $\left[\mathfrak{p}\right],\left[\mathfrak{q}\right],\left[\mathfrak{r}\right]$, that is,

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{p}\right],\left[\mathfrak{q}\right]\rangle =\langle \left[\mathfrak{p}\right],\left[\mathfrak{r}\right]\rangle =\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3).$$

(31)

The unramified cyclic cubic relative extensions of $k_{\mu }$ are among the absolutely bicyclic bicubic subfields $B_i$, $1\le i\le 10$, of the common genus field $k^{*}$ of the four components of the quartet $(k_1,\dots ,k_4)$. The unique $B_9>k_{\mu }$, $\mu \in \{2,3\}$, has the norm class group $N_{B_9/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_9\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$, and potential  fixed-point   transfer kernel

$$ker\left(T_{B_9/k_{\mu }}\right)\ge \langle \left[\mathfrak{q}\right]\rangle .$$

The unique $B_{10}>k_{\mu }$, $\mu \in \{3,4\}$, has the norm class group $N_{B_{10}/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{10}\right)\right)=\langle \left[\mathfrak{r}\right]\rangle$, and potential  fixed-point   transfer kernel

$$ker\left(T_{B_{10}/k_{\mu }}\right)\ge \langle \left[\mathfrak{r}\right]\rangle .$$

The unique $B_8>k_{\mu }$, $\mu \in \{2,4\}$, has norm class group $N_{B_8/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_8\right)\right)=\langle \left[{\mathfrak{qr}}^n\right]\rangle$, and the potential  fixed-point   transfer kernel

$$ker\left(T_{B_8/k_{\mu }}\right)\ge \langle \left[{\mathfrak{q}}^n\mathfrak{r}\right]\rangle .$$

The remaining $B_i>k_{\mu }$, $i\in \{5,6,7\}$, more precisely, $i=7$ for $\mu =2$, $i=5$ for $\mu =3$, and $i=6$ for $\mu =4$, have norm class group $\langle \left[{\mathfrak{q}}^2{\mathfrak{r}}^{-n}\right]\rangle$ and a hidden or explicit  transposition   transfer kernel, with respect to the corresponding μ.

Proof. 

As mentioned in the proof of Lemma 4, q splits in $B_9$, r splits in $B_{10}$, and p splits in $B_8$, where $\left[\mathfrak{p}\right]=\left[{\mathfrak{qr}}^n\right]$, according to (29), independently of $n\in \{1,2\}$.

• Now, we use Corollary 3 and Proposition 2.
• Since $\mathfrak{q}$ is principal in $k_q$, $\left[\mathfrak{q}\right]$ capitulates in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$.
• Since $\mathfrak{r}$ is principal in $k_r$, $\left[\mathfrak{r}\right]$ capitulates in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$ and $B_{10}=k_3{k}_4{k}_r{k}_{pq}$.
• Since ${\mathfrak{q}}^n\mathfrak{r}$ is principal in $k_{qr}$, $\left[{\mathfrak{q}}^n\mathfrak{r}\right]$ capitulates in $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$ and $B_8=k_2{k}_4{k}_p{k}_{qr}$. □

In terms of capitulation targets in Corollary 2, Proposition 5 and parts of its proof are now summarized in

Table 4. Norm class groups and minimal transfer kernels with $n=2$ for Graph I.1.

Base	${\mathit{k}}_2$				${\mathit{k}}_3$				${\mathit{k}}_4$
Ext	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$	${\mathit{B}}_4$	${\mathit{B}}_6$	${\mathit{B}}_8$	${\mathit{B}}_{10}$
NCG	$\mathfrak{r}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$
TK	$\mathfrak{qr}$	$\mathfrak{r}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$
$\varkappa$	$\mathbf{2}$	$\mathbf{1}$	3	4	$\mathbf{2}$	$\mathbf{1}$	3	4	$\mathbf{2}$	$\mathbf{1}$	3	4

for the minimal transfer kernel type (mTKT) and $n=2$, with transposition in boldface font. This essential new perspective admits progress beyond Ayadi’s work [2].

Theorem 10

(Second 3-class group for $\mathrm{I}.1$). Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$ belonging to Graph 1 of Category $\mathrm{I}$, that is, ${[p,q,r]}_3=\{p,q,r;\delta \equiv 0\left(\mathrm{mod}3\right)\}$. Without loss of generality, suppose that ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$, for $\mu =2,3,4$, and ${\varrho }_3\left(k_1\right)=3$.

Then the  minimal transfer kernel type   (mTKT) of $k_{\mu }$, $2\le \mu \le 4$, is ${\varkappa }_0=\left(4231\right)$, type $\mathrm{G}.16$, and other possible capitulation types in ascending partial order ${\varkappa }_0<\varkappa <{\varkappa }^{\prime },{\varkappa }^{″}<{\varkappa }^{‴}$ are $\varkappa =\left(0231\right)$, type $\mathrm{c}.21$, ${\varkappa }^{\prime }=\left(0001\right)$, type $\mathrm{a}.3$, ${\varkappa }^{″}=\left(0200\right)$, type $\mathrm{a}.2$, and ${\varkappa }^{‴}=\left(0000\right)$, type $\mathrm{a}.1$.

In order to identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $2\le \mu \le 4$, let the  principal factor   of $k_1$ be $A\left(k_1\right)=p^{e_1}{q}^{e_2}{r}^{e_3}$, and define $\mathcal{D}:=\#\{1\le i\le 3\mid e_i\ne 0\}$. In the  regular   situation where ${\mathrm{Cl}}_3\left(k_1\right)\simeq (3,3,3)$ is elementary tricyclic, we have

$$\mathfrak{M}\simeq \left\{\begin{array}{cc}\langle 81,8\rangle ,\alpha =[11,11,11,21],\varkappa =\left(0001\right)\hfill & \mathit{once}\mathit{if}\mathcal{D}=2,\mathcal{N}=1,\hfill \\ {\langle 81,10\rangle }^2,\alpha =[11,21,11,11],\varkappa =\left(0200\right)\hfill & \mathit{twice}\mathit{if}\mathcal{D}=2,\mathcal{N}=1,\hfill \\ \langle 243,25\rangle ,\alpha =[11,11,11,22],\varkappa =\left(0001\right)\hfill & \mathit{once}\mathit{if}\mathcal{D}=3,\mathcal{N}=1,\hfill \\ {\langle 243,28..30\rangle }^2,\alpha =[21,11,11,11],\varkappa =\left(0000\right)\hfill & \mathit{twice}\mathit{if}\mathcal{D}=3,\mathcal{N}=0,\hfill \end{array}\right.$$

(32)

where $\mathcal{N}:=\#\{1\le j\le 10\mid k_{\mu }<B_j,I_j=27\}$. In the  (super-)singular   situation where $81\mid h\left(k_1\right)$ and ${\mathrm{Cl}}_3\left(k_1\right)$ is non-elementary tricyclic, we have $\mathfrak{M}\simeq$

$$\left\{\begin{array}{cc}{\langle 243,8\rangle }^3,\alpha =[21,21,21,21],\varkappa =\left(0231\right)\hfill & \mathit{if}{h}_3\left(k_1\right)=81,\mathcal{D}=2,\mathcal{N}=3,\hfill \\ {\langle 729,54\rangle }^3,\alpha =[22,21,21,21],\varkappa =\left(0231\right)\hfill & \mathit{if}{h}_3\left(k_1\right)=81,\mathcal{D}=3,\mathcal{N}=3,\hfill \\ {\langle 2187,301|305\rangle }^3,\alpha =[32,21,21,21],\varkappa =\left(4231\right)\hfill & \mathit{if}{h}_3\left(k_1\right)=81,\mathcal{D}=3,\mathcal{N}=4,\hfill \\ {\langle 2187,303\rangle }^3,\alpha =[32,21,21,21],\varkappa =\left(0231\right)\hfill & \mathit{if}{h}_3\left(k_1\right)=243,\mathcal{D}=3,\mathcal{N}=3.\hfill \end{array}\right.$$

(33)

With the exception of the last three rows, the 3-class field tower has the group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$, ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$, since $d_2\left(\mathfrak{M}\right)\le 4$. For the last three rows, the tower length is $2\le {\mathsf{\ell }}_3\left(k_{\mu }\right)\le 3$ [16]. (See the associated descendant tree ${\mathcal{T}}^2\langle 243,8\rangle$ in [8] (Fig. 6.4, p. 63)).

Proof. 

In the non-uniform regular situations, we have $r_j=2$, $h_3\left(B_j\right)=I_j\in \{9,27\}$ for the tame bicyclic bicubic fields $j\in \{2,3,4,8,9,10\}$. Now, we use Lemmas 2 and 4.

If $\mathcal{D}=3$, then all tame indices of subfield units $I_j=9$ are minimal, and the ranks of wild bicyclic bicubic fields are $r_j=2$ for $j=5,6,7$, but non-uniform indices two times $I_5=I_6=9$, i.e., $\mathcal{N}=0$, and one time $I_7=27$, i.e., $\mathcal{N}=1$, corresponding to total capitulation twice and non-fixed-point capitulation once (due to a hidden transposition). According to Theorem 9, the common ${\alpha }_2=\left(21\right)$, and Corollary 4, the Artin pattern $\alpha =[21,11,11,11]$ and $\varkappa =\left(0000\right)$ determines three possible groups $\langle 243,28..30\rangle$, and $\alpha =[11,11,11,22]$, $\varkappa =\left(0001\right)$ uniquely leads to $\langle 243,25\rangle$.

If $\mathcal{D}=2$, then tame indices of subfield units $I_j$ are non-uniform, two times $I_j=9$, for $j=2,4,9,10$, and one time $I_j=27$, for $j=3,8$, the latter corresponding to fixed-point capitulation twice, $j=8$ over $\mu =2,4$, and non-fixed-point capitulation once, $j=3$ over $\mu =3$. So, $\mathcal{N}=1$, since the ranks of wild bicyclic bicubic fields are $r_j=3$ with uniform index $I_j=3$ for $j=5,6,7$, corresponding to a total capitulation. The Artin pattern $\alpha =[11,21,11,11]$, $\varkappa =\left(0200\right)$ uniquely determines the group $\langle 81,10\rangle$, and $\alpha =[11,11,11,21]$, $\varkappa =\left(0001\right)$ uniquely leads to $\langle 81,8\rangle$.

In the uniform singular situation with TKT $\mathrm{c}.21$, $\varkappa =\left(0231\right)$, $\mathcal{N}=3$, the ATIs decide about the group: $\alpha =[21,21,21,21]$ uniquely identifies $\langle 243,8\rangle$, $\alpha =[22,21,21,21]$ leads to $\langle 729,54\rangle$, and in the super-singular situation, $\alpha =[32,21,21,22]$ leads to $\langle 2187,303\rangle$. In contrast, for TKT $\mathrm{G}.16$, $\varkappa =\left(4231\right)$, $\mathcal{N}=4$, the ATIs $\alpha =[32,21,21,21]$ lead to $\langle 2187,301|305\rangle$.

The regular groups are of maximal class, which guarantees length ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$ of the tower. The annihilator ideal of $\langle 243,8\rangle$ is $\mathfrak{L}$, which enforces ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$, according to Scholz and Taussky [23]. The (super-)singular groups $\langle 729,54\rangle$ and $\langle 2187,303\rangle$ have non-metabelian descendants. Although they satisfy the bound $d_2\left(\mathfrak{G}\right)\le 4$ for the relation rank, a tower with three stages could only be excluded by means of computationally expensive invariants ${\alpha }^{\left(2\right)}$ of second order. □

Corollary 5.

(Non-uniformity of the sub-triplet for $\mathrm{I}.1$.)  Only two components of the sub-triplet with 3-rank two share a common capitulation type $\varkappa \left(k_{\lambda }\right)\sim \varkappa \left(k_{\mu }\right)$, common abelian-type invariants $\alpha \left(k_{\lambda }\right)\sim \alpha \left(k_{\mu }\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\lambda }\right)/k_{\lambda })\simeq \mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$. The invariants of the third component $k_{\nu }$ differ   in the  regular   situation ${\mathrm{Cl}}_3\left(k_1\right)\simeq (3,3,3)$; however, they  agree   in the  (super-)singular   situation $81\mid h\left(k_1\right)$. Here, $\{\lambda ,\mu ,\nu \}=\{2,3,4\}$.

Proof. 

This is an immediate consequence of Theorem 10. □

Example 1.

Examples 1–9 are supplemented by [8] (Tbl. 6.4–6.21, pp. 49–67). The prototypes for Graph $\mathrm{I}.1$, that is, the minimal conductors for each scenario in Theorem 10, are as follows.

There are  regular   cases: $c=4977$ with symbol $\{9,7,79\}$ and, non-uniformly, $\mathfrak{G}=\mathfrak{M}=\langle 243,25\rangle$, ${\langle 243,28..30\rangle }^2$ (Corollary 4); $c=11349$ with symbol $\{9,13,97\}$ and, non-uniformly, $\mathfrak{G}=\mathfrak{M}=\langle 81,8\rangle$, ${\langle 81,10\rangle }^2$. For 38 examples see [8] (Tbl. 6.18–6.19, pp. 64–65).

Further,  singular   cases: $c=28791$ with symbol $\{9,7,457\}$ and $\mathfrak{M}={\langle 729,54\rangle }^3$; $c=38727$ with symbol $\{9,13,331\}$ and $\mathfrak{G}=\mathfrak{M}={\langle 243,8\rangle }^3$; and, with  considerable statistic delay, there occurred $c=417807$ with ordinal number 189, symbol $\{9,13,3571\}$ and $\mathfrak{M}={\langle 2187,301|305\rangle }^3$.

And  super-singular   cases: $c=67347$ with symbol $\{9,7,1069\}$ and $\mathfrak{M}={\langle 2187,303\rangle }^3$; $c=436267$ with symbol $\{13,37,907\}$ and $\mathfrak{M}={(\langle 6561,2050\rangle -\#1;3|5)}^3$.

In

Table 5. Prototypes for Graph I.1.

No.	c	$\mathit{p},\mathit{q},\mathit{r}$	v	$\mathit{x},\mathit{y},\mathit{z};\mathsf{\ell },\mathit{m},\mathit{n}$	Capitulation Type	$\mathfrak{M}$	${\mathsf{\ell }}_3\left(\mathit{k}\right)$
1	4977	$9,7,79$	3	$2,1,1;1,1,2$	$\mathrm{a}.3,\mathrm{a}.1$	$\langle 243,25\rangle ,{\langle 243,28..30\rangle }^2$	$=2$
3	$\mathrm{11,349}$	$9,13,97$	3	$0,1,1;2,1,1$	$\mathrm{a}.3,\mathrm{a}.2$	$\langle 81,8\rangle ,{\langle 81,10\rangle }^2$	$=2$
10	$\mathrm{28,791}$	$9,7,457$	4	$2,1,1;1,1,2$	$\mathrm{c}.21$	${\langle 729,54\rangle }^3$	$\ge 2$
14	$\mathrm{38,727}$	$9,13,331$	4	$1,0,1;2,1,1$	$\mathrm{c}.21$	${\langle 243,8\rangle }^3$	$=2$
27	$\mathrm{67,347}$	$9,7,1069$	5	$2,1,1;1,1,2$	$\mathrm{c}.21$	${\langle 2187,303\rangle }^3$	$\ge 2$
189	$\mathrm{417,807}$	$9,13,3571$	4	$2,2,1;2,2,2$	$\mathrm{G}.16$	${\langle 2187,301|305\rangle }^3$	$\ge 2$
198	$\mathrm{436,267}$	$13,37,907$	6	$1,1,1;2,2,2$	$\mathrm{G}.16$	${(R-\#1;3|5)}^3$	$\ge 2$

, we summarize the prototypes of graph $\mathrm{I}.1$. The data comprise ordinal number No.; conductor c of k; combined cubic residue symbol ${[p,q,r]}_3$; regularity or (super-)singularity expressed by 3-valuation $v=v_3(\#\mathrm{Cl}\left(k_1\right))$ of class number of critical field $k_1$; critical exponents $x,y,z$ in principal factor $A\left(k_1\right)=p^x{q}^y{r}^z$ and $\mathsf{\ell },m,n$ in $A\left(k_{pq}\right)=p^{\mathsf{\ell }}q$, $A\left(k_{pr}\right)=p^{m}r$, $A\left(k_{qr}\right)=q^{n}r$; capitulation type of k; second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ of k; and length ${\mathsf{\ell }}_3\left(k\right)$ of 3-class field tower of k.

We put $R:=\langle 6561,2050\rangle$ for abbreviation.

6.2. Category I, Graph 2

Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 2 of Category $\mathrm{I}$ with combined cubic residue symbol ${[p,q,r]}_3=\{q\leftarrow p\to r\}$.

Lemma 5.

(3-class ranks of components for $\mathrm{I}.2$.)  Under the normalizing assumptions that q splits in $k_{pr}$ and r splits in $k_{pq}$, precisely the three components $k_2$, $k_3$, $k_4$ of the quartet have the elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$, $\mu =2,3,4$, of rank 2, whereas the remaining component has the 3-class rank ${\varrho }_3\left(k_1\right)=3$. Thus, the  tame   condition $9\mid h_3\left(B_j\right)=(U_j:V_j)\in \{9,27\}$, $r_j=2$, is satisfied for the bicyclic bicubic fields $B_j$ with $j\in \{2,3,4,8,9,10\}$.

Proof. 

p is universally repelling $\{q\leftarrow p\to r\}$. Since $p\to r$, p splits in $k_r$. Since $q\leftarrow p$, p splits in $k_q$. Thus, p also splits in $k_{qr}$ and ${\tilde{k}}_{qr}$. By the normalizing assumptions that q splits in $k_{pr}$ and r splits in $k_{pq}$, the primes $p,q,r$ share the common decomposition type $(e,f,g)=(3,1,3)$ in the bicyclic bicubic field $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, which implies that ${\varrho }_3\left(k_1\right)=3$, according to [2] (Prop. 4.4, pp. 43–44). Finally, none among $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$, $B_{10}=k_3{k}_4{k}_r{k}_{pq}$ contains $k_1$. □

Proposition 6.

(Sub-triplet with 3-rank two for $\mathrm{I}.2$.)  For fixed $\mu \in \{2,3,4\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$, then the 3-class group of $k_{\mu }$ is generated by the non-trivial classes $\left[\mathfrak{q}\right],\left[\mathfrak{r}\right]$, that is,

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3).$$

(34)

The unramified cyclic cubic relative extensions of $k_{\mu }$ are among the absolutely bicyclic bicubic fields $B_i$, $1\le i\le 10$.

In terms of  decisive principal factors $A\left(k_1\right)=p^x{q}^y{r}^z$, $x,y,z\in \{0,1,2\}$, and $A\left(k_{qr}\right)=q{r}^n$, $n\in \{1,2\}$, the ranks of principal factor matrices of  wild   bicyclic bicubic fields are

$$r_5=3\mathit{iff}-z\not\equiv ny\left(\mathrm{mod}3\right),r_6=3\mathit{iff}z\ne 0\mathit{iff}r\mid A\left(k_1\right),r_7=3\mathit{iff}y\ne 0\mathit{iff}q\mid A\left(k_1\right).$$

(35)

The field $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$ has the norm class group $N_{B_2/k_2}\left({\mathrm{Cl}}_3\left(B_2\right)\right)=\langle \left[\mathfrak{r}\right]\rangle$, and the transfer kernel

$$ker\left(T_{B_2/k_2}\right)\ge \langle \left[{\mathfrak{q}}^2{\mathfrak{r}}^n\right]\rangle .$$

The field $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$ has the norm class group $N_{B_4/k_4}\left({\mathrm{Cl}}_3\left(B_4\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$, and the transfer kernel

$$ker\left(T_{B_4/k_4}\right)\ge \langle \left[{\mathfrak{q}}^2{\mathfrak{r}}^n\right]\rangle .$$

The field $B_9=k_2{k}_3{k}_q{k}_{pr}$, which contains $k_2$ and $k_3$, has the norm class group $N_{B_9/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_9\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$, for $\mu =2,3$, and the possible  fixed-point   transfer kernel

$$ker\left(T_{B_9/k_{\mu }}\right)\ge \langle \left[\mathfrak{q}\right]\rangle .$$

(36)

The field $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, which contains $k_3$ and $k_4$, has the norm class group $N_{B_{10}/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{10}\right)\right)=\langle \left[\mathfrak{r}\right]\rangle$, for $\mu =3,4$, and the possible  fixed-point   transfer kernel

$$ker\left(T_{B_{10}/k_{\mu }}\right)\ge \langle \left[\mathfrak{r}\right]\rangle .$$

(37)

The remaining two $B_i>k_{\mu }$, $i\in \{3,5,6,7,8\}$, more precisely, $i\in \{7,8\}$ for $\mu =2$, $i\in \{3,5\}$ for $\mu =3$, and $i\in \{6,8\}$ for $\mu =4$, have the norm class group $\langle \left[\mathfrak{qr}\right]\rangle$ or $\langle \left[{\mathfrak{qr}}^2\right]\rangle$. Among them, the tame extensions $B_i>k_{\mu }$ with either $i=\mu =3$ or $i=8$, $\mu =2,4$, have the  partial   transfer kernel

$$ker\left(T_{B_i/k_{\mu }}\right)=\langle \left[\mathfrak{qr}\right]\rangle$$

(38)

of order 3, giving rise to either a transposition or a fixed point.

Proof. 

Since $p\to r$, two principal factors are $A\left(k_{pr}\right)=A\left({\tilde{k}}_{pr}\right)=p$; and since $q\leftarrow p$, two further principal factors are $A\left(k_{pq}\right)=A\left({\tilde{k}}_{pq}\right)=p$, by Proposition 3. Since p is universally repelling $\{q\leftarrow p\to r\}$, three further principal factors are $A\left(k_{\mu }\right)=p$ for $2\le \mu \le 4$, by Proposition 4.

Thus, $\left[\mathfrak{p}\right]=1$ is trivial, and the non-trivial classes $\left[\mathfrak{q}\right],\left[\mathfrak{r}\right]$ generate ${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3)$.

Since q splits in $k_{pr}$, it also splits in $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$.

Since r splits in $k_{pq}$, it also splits in $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_{10}=k_3{k}_4{k}_r{k}_{pq}$.

Since the tame condition $9\mid h_3\left(B_j\right)=(U_j:V_j)$ is satisfied for $j\in \{2,3,4,8,9,10\}$, the rank of the corresponding principal factor matrix $M_j$ must be $r_j=2$. This can also be verified directly and has no further consequences.

We propose the principal factors $A\left(k_1\right)=p^x{q}^y{r}^z$ and $A\left(k_{qr}\right)=q{r}^n$, $A\left({\tilde{k}}_{qr}\right)=q^2{r}^n$ with $n\in \{1,2\}$. For each wild bicyclic bicubic field $B_j$, $j\in \{5,6,7\}$, the rank $r_j$ is now calculated with row operations on the associated principal factor matrices $M_j$:

$M_5=\left(\begin{array}{ccc}x& y& z\\ 1& 0& 0\\ 1& 0& 0\\ 0& 2& n\end{array}\right)$, $M_6=\left(\begin{array}{ccc}x& y& z\\ 1& 0& 0\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$, $M_7=\left(\begin{array}{ccc}x& y& z\\ 1& 0& 0\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$.

For $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $M_5$ leads to the decisive pivot element $z+ny$ in the last column. So, $r_5=3$ iff $-z\not\equiv ny$ modulo 3. For $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $M_6$ leads to z. So, $r_6=3$ iff $z\ne 0$. For $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ leads to y in the middle column. So, $r_7=3$ iff $y\ne 0$.

Since $\mathfrak{q}$ is principal in $k_q$, $\left[\mathfrak{q}\right]$ capitulates in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$, and since $\mathfrak{r}$ is principal in $k_r$, $\left[\mathfrak{r}\right]$ capitulates in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$ and $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, by Corollary 3.

Since ${\mathfrak{qr}}^n$ is principal in $k_{qr}$, $\left[{\mathfrak{qr}}^n\right]$ capitulates in $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$, and since ${\mathfrak{q}}^2{\mathfrak{r}}^n$ is principal in ${\tilde{k}}_{qr}$, $\left[{\mathfrak{q}}^2{\mathfrak{r}}^n\right]$ capitulates in $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, and $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, by Proposition 2.

In each case, the minimal subfield unit index $(U_j:V_j)=3$ for $r_j=3$ corresponds to the maximal unit norm index $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3$, associated with a total transfer kernel $\#ker\left(T_{B_j/k_{\mu }}\right)=9$, by Lemma 2.

The minimal unit norm index $(U\left(k_{\mu }\right):N_{B_8/k_{\mu }}\left(U_8\right))=1$, associated with the partial transfer kernel $ker\left(T_{B_8/k_{\mu }}\right)=\langle \left[\mathfrak{qr}\right]\rangle$, for $\mu =2,4$, corresponds to the tame maximal subfield unit index $h_3\left(B_8\right)=(U_8:V_8)=27$, giving rise to type invariants ${\mathrm{Cl}}_3\left(B_8\right)\simeq (9,3)$. □

Using Corollary 2, Proposition 6 and parts of its proof are now summarized in

Table 6. Norm class groups and minimal transfer kernels with $n=1$ for Graph I.2.

Base	${\mathit{k}}_2$				${\mathit{k}}_3$				${\mathit{k}}_4$
Ext	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$	${\mathit{B}}_4$	${\mathit{B}}_6$	${\mathit{B}}_8$	${\mathit{B}}_{10}$
NCG	$\mathfrak{r}$	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{r}$
TK	${\mathfrak{qr}}^2$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{r}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{qr}$	$\mathfrak{r}$
$\varkappa$	$\mathbf{2}$	$\mathbf{1}$	3	4	$\mathbf{2}$	$\mathbf{1}$	3	4	$\mathbf{2}$	$\mathbf{1}$	3	4

for the minimal transfer kernel type (mTKT) and $n=1$, with transposition in boldface font.

Theorem 11

(Second 3-class group for $\mathrm{I}.2$). To identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $2\le \mu \le 4$, let the principal factors of $k_1$, $k_{qr}$, and ${\tilde{k}}_{qr}$, respectively, be $A\left(k_1\right)=p^x{q}^y{r}^z$, $x,y,z\in \{0,1,2\}$, $A\left(k_{qr}\right)=q{r}^n$, and $A\left({\tilde{k}}_{qr}\right)=q^2{r}^n$, $n\in \{1,2\}$, respectively, and additionally assume the regular situation where ${\mathrm{Cl}}_3\left(k_1\right)\simeq (3,3,3)$.

Then, the  minimal transfer kernel type   (mTKT) ${\varkappa }_0$ of $k_{\mu }$, $1\le \mu \le 4$, and other possible capitulation types in ascending partial order ${\varkappa }_0<\varkappa <{\varkappa }^{\prime },{\varkappa }^{″}$, ending in two non-comparable types, are ${\varkappa }_0=\left(2134\right)$, type $\mathrm{G}.16$, $\varkappa =\left(0134\right)$, type $\mathrm{c}.21$, ${\varkappa }^{\prime }=\left(0004\right)$, type $\mathrm{a}.2$, ${\varkappa }^{″}=\left(0100\right)$, type $\mathrm{a}.3$, and the second 3-class group is $\mathfrak{M}\simeq$

$$\left\{\begin{array}{cc}\langle 81,8\rangle ,\alpha =[11,21,11,11],\varkappa =\left(0100\right)\hfill & \mathit{once}\mathit{if}y\ne 0,z\ne 0,\mathcal{N}=1,\hfill \\ \langle 81,10\rangle ,\alpha =[11,11,11,21],\varkappa =\left(0004\right)\hfill & \mathit{twice}\mathit{if}y\ne 0,z\ne 0,\mathcal{N}=1,\hfill \\ \langle 243,8\rangle ,\alpha =[21,21,21,21],\varkappa =\left(0134\right)\hfill & \mathit{if}y=z=0,\mathcal{N}=3,\hfill \\ \langle 729,52\rangle ,\alpha =[22,21,21,21],\varkappa =\left(2134\right)\hfill & \mathit{if}y=z=0,\mathcal{N}=4,\hfill \end{array}\right.$$

(39)

where $\mathcal{N}:=\#\{1\le j\le 10\mid k_{\mu }<B_j,I_j=27\}$. Only for the leading three rows, the 3-class field tower has certainly the group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$ and length ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$, otherwise length ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$ cannot be excluded although $d_2\left(\mathfrak{M}\right)\le 4$.

Proof. 

Let $\mu \in \{2,3,4\}$.

The first scenario, $y\ne 0$, $z\ne 0$, and $-z\not\equiv ny$ modulo 3 is equivalent to $\mathcal{N}=1$, since $r_j=3$, $(U_j:V_j)=3$, $h_3\left(B_j\right)=\frac{1}{3}{h}_3\left(k_1\right)=9$, for the wild $j=5,6,7$, and $h_3\left(B_j\right)=(U_j:V_j)=9$, for the tame $j=2,4,9,10$, whereas the distinguished tame $j=3,8$ have $h_3\left(B_j\right)=(U_j:V_j)=27$. This gives rise to the Artin pattern of either $\alpha =[11,11,11,21]$, $\varkappa =\left(0004\right)$, for $j=8$, $\mu =2,4$ (twice with fixed point), characteristic for $\langle 81,10\rangle$, or $\alpha =[11,21,11,11]$, $\varkappa =\left(0100\right)$, for $j=\mu =3$ (only once with a non-fixed point, due to a hidden transposition), characteristic for $\langle 81,8\rangle$.

The other two scenarios share $y=z=0$, and thus also $-z=ny$, independently of n, which implies $r_j=2$, $(U_j:V_j)\in \{9,27\}$, for $j=5,6,7$, and $h_3\left(B_j\right)=(U_j:V_j)=27$, for the tame $j=2,4,9,10$, producing two fixed points at $B_9$ and $B_{10}$.

The second scenario with $\mathcal{N}=3$ is supplemented by $(U_j:V_j)=9$, $h_3\left(B_j\right)=h_3\left(k_1\right)=27$, for $j=5,6,7$, and total capitulation, $\#ker\left(T_{B_j/k_{\mu }}\right)=9$, for $\mu =3,4,2$. This gives rise to $\alpha =[21,21,21,21]$, $\varkappa =\left(0134\right)$, characteristic for $\langle 243,8\rangle$ with annihilator ideal $\mathfrak{L}$ in the sense of Scholz and Taussky [23].

The third scenario with $\mathcal{N}=4$ is supplemented by $(U_j:V_j)=27$, $h_3\left(B_j\right)=3h_3\left(k_1\right)=81$, for $j=5,6,7$ and partial non-fixed-point capitulation. This gives rise to $\alpha =[22,21,21,21]$, $\varkappa =\left(2134\right)$, characteristic for $\langle 729,52\rangle$ with non-metabelian descendants. Here, the hidden transposition becomes explicit, between either $B_2$, $B_7$ or $B_3$, $B_5$ or $B_4$, $B_6$. □

Corollary 6

(Non-uniformity of the sub-triplet for $\mathrm{I}.2$). The components of the sub-triplet with 3-rank two share a common capitulation type $\varkappa \left(k_{\mu }\right)$, common abelian-type invariants $\alpha \left(k_{\mu }\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, for $\mu =2,3,4$, only if $y=z=0$, $\mathcal{N}=3,4$. For $y\ne 0$, $z\ne 0$, $\mathcal{N}=1$, however, only two fields $k_2$ and $k_4$ share common invariants, whereas $k_3$ has different $\varkappa \left(k_3\right)$ and different $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_3\right)/k_3)$.

Proof. 

This follows immediately from Theorem 11, whereas Table 6 with the minimal transfer kernel type ${\varkappa }_0=\left(2134\right)$ only shows the uniform situation, which can become non-uniform by superposition with total transfer kernels, when $\mathcal{N}=1$. □

Example 2.

For 60 examples see [8] (Tbl. 6.20–6.21, pp. 66–67). Prototypes for Graph I.2, i.e., minimal conductors for each scenario in Theorem 11, have been found for each $\mathcal{N}\in \{1,3,4\}$.

Some are  regular: $c=8001$ with symbol $\{9\leftarrow 127\to 7\}$ and, non-uniformly, once $\mathfrak{G}=\mathfrak{M}=\langle 81,8\rangle$ but twice ${\langle 81,10\rangle }^2$; $c=21049$ with symbol $\{7\leftarrow 97\to 31\}$ and uniformly three times $\mathfrak{G}=\mathfrak{M}={\langle 243,8\rangle }^3$; and $c=59031$ with symbol $\{9\leftarrow 937\to 7\}$ and $\mathfrak{M}={\langle 729,52\rangle }^3$.

Others are  singular: $c=7657$ with symbol $\{13\leftarrow 31\to 19\}$ and $\mathfrak{G}=\mathfrak{M}={\langle 243,8\rangle }^3$; and $c=48393$ with symbol $\{9\leftarrow 19\to 283\}$ and $\mathfrak{M}={\langle 2187,301|305\rangle }^3$.

The groups of order $\ge 729$ with the transfer kernel type $\mathrm{G}.16$ have non-metabelian extensions.

In

Table 7. Prototypes for Graph I.2.

No.	c	$\mathit{q}\leftarrow \mathit{p}\to \mathit{r}$	v	$\mathit{y},\mathit{z};\mathit{n}$	Capitulation Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	7657	$13\leftarrow 31\to 19$	4	$1,1;2$	$\mathrm{c}.21$	${\langle 243,8\rangle }^3$	$=2$
2	8001	$9\leftarrow 127\to 7$	3	$1,2;1$	$\mathrm{a}.3,\mathrm{a}.2$	$\langle 81,8\rangle ,{\langle 81,10\rangle }^2$	$=2$
12	$\mathrm{21,049}$	$7\leftarrow 97\to 31$	3	$0,0;1$	$\mathrm{c}.21$	${\langle 243,8\rangle }^3$	$=2$
27	$\mathrm{48,393}$	$9\leftarrow 19\to 283$	4	$0,0;2$	$\mathrm{G}.16$	${\langle 2187,301|305\rangle }^3$	$\ge 2$
33	$\mathrm{59,031}$	$9\leftarrow 937\to 7$	3	$0,0;1$	$\mathrm{G}.16$	${\langle 729,52\rangle }^3$	$\ge 2$

, we summarize the prototypes of Graph $\mathrm{I}.2$ in the same way as in Table 5, except that two critical exponents $y,z$ in the principal factor $A\left(k_1\right)=p^x{q}^y{r}^z$ and n in $A\left(k_{qr}\right)=q{r}^n$ are sufficient.

6.3. Category II, Graph 1

Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 1 of Category $\mathrm{II}$ with combined cubic residue symbol ${[p,q,r]}_3=\{p\to q\leftarrow r\}$.

Lemma 6

(3-class ranks of components for $\mathrm{II}.1$). Under the normalizing assumption that q splits in ${\tilde{k}}_{pr}$, precisely the two components $k_2$ and $k_3$ of the quartet have the elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k_2\right)\simeq {\mathrm{Cl}}_3\left(k_3\right)\simeq (3,3)$ of rank 2, whereas the other two components have the 3-class rank ${\varrho }_3\left(k_1\right)={\varrho }_3\left(k_4\right)=3$. Thus, the tame condition $9\mid h_3\left(B_j\right)=(U_j:V_j)\in \{9,27\}$, $r_j=2$, is only satisfied for the bicyclic bicubic fields $B_j$ with $j\in \{2,3,9\}$.

Proof. 

Since $p\to q$, p splits in $k_q$. Since $q\leftarrow r$, r splits in $k_q$, and also splits in $B_9=k_2{k}_3{k}_q{k}_{pr}$. By the normalizing assumption that q splits in ${\tilde{k}}_{pr}$, it also splits in $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$ and $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$. The primes $p,q,r$ share the common decomposition type $(e,f,g)=(3,1,3)$ in the bicyclic bicubic field $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, which implies that ${\varrho }_3\left(k_1\right)={\varrho }_3\left(k_4\right)=3$, according to ([2] Prop. 4.4, pp. 43–44). Finally, only $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$ do not contain $k_1$, $k_4$. □

Proposition 7

(Sub-doublet with 3-rank two for $\mathrm{II}.1$). For fixed $\mu \in \{2,3\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$, then the 3-class group of $k_{\mu }$ is generated by the non-trivial classes $\left[\mathfrak{q}\right],\left[\mathfrak{r}\right]$, that is,

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3).$$

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The unramified cyclic cubic relative extensions of $k_{\mu }$ are among the absolutely bicyclic bicubic fields $B_i$, $1\le i\le 10$. The unique $B_{\mu }$, $\mu \in \{2,3\}$, which only contains $k_{\mu }$, has the norm class group $N_{B_{\mu }/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{\mu }\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$, the transfer kernel

$$ker\left(T_{B_{\mu }/k_{\mu }}\right)\ge \langle \left[\mathfrak{r}\right]\rangle$$

and the 3-class group ${\mathrm{Cl}}_3\left(B_{\mu }\right)=\langle \left[{\mathfrak{Q}}_1\right],\left[{\mathfrak{Q}}_2\right],\left[{\mathfrak{Q}}_3\right]\rangle \ge (3,3)$, generated by the classes of the prime ideals of $B_{\mu }$ over $\mathfrak{q}{\mathcal{O}}_{B_{\mu }}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$. The unique $B_9=k_2{k}_3{k}_q{k}_{pr}$, which contains $k_2$ and $k_3$, has the norm class group $N_{B_9/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_9\right)\right)=\langle \left[\mathfrak{r}\right]\rangle$, the  cyclic   transfer kernel

$$ker\left(T_{B_9/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$$

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of order 3, and the  elementary tricyclic  3-class group ${\mathrm{Cl}}_3\left(B_9\right)=\langle \left[{\mathfrak{R}}_1\right],\left[{\mathfrak{R}}_2\right],\left[{\mathfrak{R}}_3\right]\rangle \simeq (3,3,3)$, generated by the classes of the prime ideals of $B_9$ over $\mathfrak{r}{\mathcal{O}}_{B_9}={\mathfrak{R}}_1{\mathfrak{R}}_2{\mathfrak{R}}_3$. The remaining two $B_i>k_{\mu }$, $i\in \{5,7,8,10\}$, more precisely, $i\in \{7,8\}$ for $\mu =2$, and $i\in \{5,10\}$ for $\mu =3$, have the norm class group $\langle \left[\mathfrak{qr}\right]\rangle$ or $\langle \left[{\mathfrak{qr}}^2\right]\rangle$, and the transfer kernel

$$ker\left(T_{B_i/k_{\mu }}\right)\ge \langle \left[\mathfrak{r}\right]\rangle .$$

In terms of  decisive principal factors $A\left(k_{\nu }\right)=p^{x_{\nu }}{q}^{y_{\nu }}{r}^{z_{\nu }}$ for $\nu \in \{1,4\}$, the ranks of principal factor matrices $M_i$, $i\in \{1,4,5,7,8,10\}$, of  wild   bicyclic bicubic fields are

$$r_1=r_5=r_7=3\mathit{iff}{y}_1\ne 0\mathit{iff}q\mid A\left(k_1\right)\mathit{and}{r}_4=r_8=r_{10}=3\mathit{iff}{y}_4\ne 0\mathit{iff}q\mid A\left(k_4\right).$$

(42)

Proof. 

Since $p\to q$, two principal factors are $A\left(k_{pq}\right)=A\left({\tilde{k}}_{pq}\right)=p$; since $q\leftarrow r$, two further principal factors are $A\left(k_{qr}\right)=A\left({\tilde{k}}_{qr}\right)=r$, both by Proposition 3.

Since the tame condition $9\mid h_3\left(B_j\right)=(U_j:V_j)$ is satisfied for $j\in \{2,3,9\}$, the rank of the corresponding principal factor matrix $M_j$ must be $r_2=r_3=r_9=2$. We propose principal factors $A\left(k_{\mu }\right)=p^{x_{\mu }}{q}^{y_{\mu }}{r}^{z_{\mu }}$, for all $1\le \mu \le 4$, and $A\left(k_{pr}\right)=p{r}^{\mathsf{\ell }}$, $A\left({\tilde{k}}_{pr}\right)=p^2{r}^{\mathsf{\ell }}$ with $\mathsf{\ell }\in \{1,2\}$.

For each bicyclic bicubic field $B_j$, the rank $r_j$ is calculated with row operations on the associated principal factor matrices $M_j$:

$M_2=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ 1& 0& 0\\ 2& 0& \mathsf{\ell }\\ 0& 0& 1\end{array}\right)$, $M_3=\left(\begin{array}{ccc}{x}_3& y_3& z_3\\ 1& 0& 0\\ 2& 0& \mathsf{\ell }\\ 0& 0& 1\end{array}\right)$, $M_9=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ x_3& y_3& z_3\\ 0& 1& 0\\ 1& 0& \mathsf{\ell }\end{array}\right)$.

For $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $M_2$ leads to the decisive pivot element $y_2$ in the middle column, and similarly, for $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $M_3$ leads to $y_3$. So, $r_2=r_3=2$ enforces $y_2=y_3=0$, i.e., $q\nmid A\left(k_2\right)$, $q\nmid A\left(k_3\right)$. However, for $B_9=k_2{k}_3{k}_q{k}_{pr}$, $M_9$ leads to $z_2-\mathsf{\ell }{x}_2$ and $z_3-\mathsf{\ell }{x}_3$. So, $r_9=2$ enforces $z_2\equiv \mathsf{\ell }{x}_2$ and $z_3\equiv \mathsf{\ell }{x}_3$ modulo 3, i.e., $A\left(k_2\right)=A\left(k_3\right)=p{r}^{\mathsf{\ell }}$.

For every wild bicyclic bicubic field $B_j$, $j\in \{1,4,5,6,7,8,10\}$, the rank $r_j$ is calculated by row operations on the matrices $M_j$, using $A\left(k_2\right)=A\left(k_3\right)=p{r}^{\mathsf{\ell }}$:

$M_1=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ 1& 0& 0\\ 1& 0& \mathsf{\ell }\\ 0& 0& 1\end{array}\right)$, $M_5=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ 1& 0& \mathsf{\ell }\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$, $M_7=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ 1& 0& \mathsf{\ell }\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$.

For $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $M_1$ leads to the decisive pivot element $y_1$ in the middle column; similarly, for $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $M_5$ leads to $y_1$; and similarly, for $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ leads to $y_1$. So, $r_1=r_5=r_7=3$ iff $y_1\ne 0$ iff $q\mid A\left(k_1\right)$. Next, we consider:

$M_4=\left(\begin{array}{ccc}{x}_4& y_4& z_4\\ 1& 0& 0\\ 1& 0& \mathsf{\ell }\\ 0& 0& 1\end{array}\right)$, $M_8=\left(\begin{array}{ccc}1& 0& \mathsf{\ell }\\ x_4& y_4& z_4\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$, $M_{10}=\left(\begin{array}{ccc}1& 0& \mathsf{\ell }\\ x_4& y_4& z_4\\ 0& 0& 1\\ 1& 0& 0\end{array}\right)$.

For $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_4$ leads to the decisive pivot element $y_4$ in the middle column; similarly, for $B_8=k_2{k}_4{k}_p{k}_{qr}$, $M_8$ leads to $y_4$; and similarly, for $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $M_{10}$ leads to $y_4$. So, $r_4=r_8=r_{10}=3$ iff $y_4\ne 0$ iff $q\mid A\left(k_4\right)$.

By Lemma 2, the minimal subfield unit index $(U_j:V_j)=3$ for $r_j=3$ corresponds to the maximal unit norm index $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3$, associated with a total transfer kernel $\#ker\left(T_{B_j/k_{\mu }}\right)=9$.

Since q splits in ${\tilde{k}}_{pr}$, it also splits in $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $\mathfrak{q}{\mathcal{O}}_{B_{\mu }}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$.

Since r splits in $k_q$, it also splits in $B_9=k_2{k}_3{k}_q{k}_{pr}$, $\mathfrak{r}{\mathcal{O}}_{B_9}={\mathfrak{R}}_1{\mathfrak{R}}_2{\mathfrak{R}}_3$.

Since $\mathfrak{r}$ is principal in $k_r$, $k_{qr}$, ${\tilde{k}}_{qr}$, $\left[\mathfrak{r}\right]$ capitulates in $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$, $B_{10}=k_3{k}_4{k}_r{k}_{pq}$; since $\mathfrak{q}$ is principal in $k_q$, $\left[\mathfrak{q}\right]$ capitulates in $B_9=k_2{k}_3{k}_q{k}_{pr}$ (Proposition 2). This gives a transposition, either $(2,9)$ or $(3,9)$.

The minimal unit norm index $(U\left(k_{\mu }\right):N_{B_9/k_{\mu }}\left(U_9\right))=1$, associated with the partial transfer kernel $ker\left(T_{B_9/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$, corresponds to the maximal subfield unit index $h_3\left(B_9\right)=(U_9:V_9)=27$, giving rise to the elementary tricyclic type invariants ${\mathrm{Cl}}_3\left(B_9\right)=\langle \left[{\mathfrak{R}}_1\right],\left[{\mathfrak{R}}_2\right],\left[{\mathfrak{R}}_3\right]\rangle \simeq (3,3,3)$. □

Using Corollary 2, Proposition 7 and parts of its proof are now summarized in

Table 8. Norm class groups and minimal transfer kernels for Graph II.1.

Base	${\mathit{k}}_2$				${\mathit{k}}_3$
Ext	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$
NCG	$\mathfrak{q}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$	$\mathfrak{qr}$
TK	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{r}$
$\varkappa$	$\mathbf{4}$	4	4	$\mathbf{1}$	$\mathbf{3}$	3	$\mathbf{1}$	3

with transposition in bold font.

Theorem 12

(Second 3-class group for $\mathrm{II}.1$). Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 1 of Category $\mathrm{II}$ with the combined cubic residue symbol ${[p,q,r]}_3=\{p\to q\leftarrow r\}$. Without loss of generality, suppose that q splits in ${\tilde{k}}_{pr}$, and thus, ${\mathrm{Cl}}_3\left(k_2\right)\simeq {\mathrm{Cl}}_3\left(k_3\right)\simeq (3,3)$, and ${\varrho }_3\left(k_1\right)={\varrho }_3\left(k_4\right)=3$.

Then, the  minimal transfer kernel type   (mTKT) of $k_{\mu }$, $2\le \mu \le 3$ is ${\varkappa }_0=\left(2111\right)$, type $\mathrm{H}.4$, and the other possible capitulation types in ascending order ${\varkappa }_0<{\varkappa }^{\prime }<{\varkappa }^{″}<{\varkappa }^{‴}$ are ${\varkappa }^{\prime }=\left(2110\right)$, type $\mathrm{d}.19$, ${\varkappa }^{″}=\left(2100\right)$, type $\mathrm{b}.10$, and ${\varkappa }^{‴}=\left(2000\right)$, type $\mathrm{a}.3^{*}$.

To identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $2\le \mu \le 3$, let the  decisive principal factors   of $k_{\nu }$, $\nu \in \{1,4\}$ be $A\left(k_{\nu }\right)=p^{x_{\nu }}{q}^{y_{\nu }}{r}^{z_{\nu }}$, and additionally, assume the  regular situation  where both ${\mathrm{Cl}}_3\left(k_1\right)\simeq {\mathrm{Cl}}_3\left(k_4\right)\simeq (3,3,3)$ are elementary tricyclic. Then,

$$\mathfrak{M}\simeq \left\{\begin{array}{cc}\langle 81,7\rangle ,\alpha =[111,11,11,11],\varkappa =\left(2000\right)\hfill & \mathit{if}{y}_1\ne 0,y_4\ne 0,\mathcal{N}=1,\hfill \\ \langle 729,34..39\rangle ,\alpha =[111,111,21,21],\varkappa =\left(2100\right)\hfill & \mathit{if}{y}_1=y_4=0,\mathcal{N}=2,\hfill \\ \langle 729,41\rangle ,\alpha =[111,111,22,21],\varkappa =\left(2110\right)\hfill & \mathit{if}{y}_1=y_4=0,\mathcal{N}=3,\hfill \\ \langle 6561,714..719|738..743\rangle \mathit{or}\hfill & \\ \langle 2187,65|67\rangle ,\alpha =[111,111,22,22],\varkappa =(2111)\hfill & \mathit{if}{y}_1=y_4=0,\mathcal{N}=4,\hfill \end{array}\right.$$

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where $\mathcal{N}:=\#\{1\le j\le 10\mid k_{\mu }<B_j,I_j=27\}$. Only in the leading row has the 3-class field tower warranted the group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$, with length ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$. Otherwise, although the relation rank $d_2\left(\mathfrak{M}\right)\le 4$ is always admissible, the tower length ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$ cannot be excluded.

Proof. 

We give the proof for $k_3$ with unramified cyclic cubic extensions $B_3,B_5,B_9,B_{10}$ (the proof for $k_2$ with unramified cyclic cubic extensions $B_2,B_7,B_8,B_9$ is similar). We know that the tame ranks are $r_2=r_3=r_9=2$, and thus $I_2,I_3,I_9\in \{9,27\}$, in particular, $I_9=27$, whence certainly $\mathcal{N}\ge 1$. Further, the wild ranks are $r_1=r_5=r_7=3$ iff $y_1\ne 0$, and $r_4=r_8=r_{10}=3$ iff $y_4\ne 0$.

In the regular situation where the 3-class groups of $k_1$ and $k_4$ are elementary tricyclic, tight bounds arise for the abelian quotient invariants $\alpha$ of the group $\mathfrak{M}$:

The first scenario, $y_1\ne 0$, $y_4\ne 0$ is equivalent to $\mathcal{N}=1$, $h_3\left(B_5\right)=h_3\left(B_7\right)=\frac{1}{3}{h}_3\left(k_1\right)=9$, $h_3\left(B_8\right)=h_3\left(B_{10}\right)=\frac{1}{3}{h}_3\left(k_4\right)=9$, $h_3\left(B_2\right)=I_2=h_3\left(B_3\right)=I_3=9$, $h_3\left(B_9\right)=I_9=27$, that is, $\alpha =[111,11,11,11]$ and consequently, $\varkappa =\left(2000\right)$, since $\langle 81,7\rangle$ is unique with this $\alpha$.

The other three scenarios share $y_1=y_4=0$, and an explicit transposition between $B_3$ and $B_9$, giving rise to $\varkappa =(21\ast \ast )$, and common $h_3\left(B_3\right)=I_3=27$, $\alpha =[111,111,\ast ,\ast ]$.

The second scenario with $\mathcal{N}=2$ is supplemented by

$h_3\left(B_5\right)=h_3\left(k_1\right)=27$, $h_3\left(B_{10}\right)=h_3\left(k_4\right)=27$, giving rise to $\alpha =[111,111,21,21]$, $\varkappa =\left(2100\right)$, characteristic for $\langle 729,34..39\rangle$ (Corollary 4).

The third scenario with $\mathcal{N}=3$ is supplemented by

$h_3\left(B_5\right)=3h_3\left(k_1\right)=81$, $h_3\left(B_{10}\right)=h_3\left(k_4\right)=27$, giving rise to $\alpha =[111,111,22,21]$, $\varkappa =\left(2110\right)$, characteristic for $\langle 729,41\rangle$.

The fourth scenario with $\mathcal{N}=4$ is supplemented by

$h_3\left(B_5\right)=3h_3\left(k_1\right)=81$, $h_3\left(B_{10}\right)=3h_3\left(k_4\right)=81$, giving rise to $\alpha =[111,111,22,22]$, $\varkappa =\left(2111\right)$, characteristic for either $\langle 2187,65|67\rangle$ or $\langle 6561,714..719|738..743\rangle$ with coclass $\mathrm{cc}=3$. If $d_2\left(\mathfrak{M}\right)=5$, then tower length must be ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$. For this minimal capitulation type H.4, $\varkappa =\left(2111\right)$, all transfer kernels are cyclic of order 3, and the minimal unit norm indices correspond to maximal subfield unit indices. □

Corollary 7

(Uniformity of the sub-doublet for $\mathrm{II}.1$). The components of the sub-doublet with 3-rank two share a common capitulation type $\varkappa \left(k_2\right)\sim \varkappa \left(k_3\right)$, common abelian-type invariants $\alpha \left(k_2\right)\sim \alpha \left(k_3\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_2\right)/k_2)\simeq \mathrm{Gal}({\mathrm{F}}_3^2\left(k_3\right)/k_3)$.

Proof. 

This is an immediate consequence of Theorem 12 and Table 8. □

Example 3.

For 47 examples see [8] (Tbl. 6.14–6.15, pp. 60–61). Prototypes for Graph $\mathrm{II}.1$, i.e., minimal conductors for each scenario in Theorem 12, have been detected for all $N\in \{1,2,3,4\}$.

There are  regular   cases: $c=3913$ with symbol $\{13\to 7\leftarrow 43\}$ and $\mathfrak{G}=\mathfrak{M}=\langle 81,7\rangle$; $c=22581$ with symbol $\{9\to 193\leftarrow 13\}$ and $\mathfrak{M}=\langle 729,41\rangle$; $c=25929$ with symbol $\{9\to 67\leftarrow 43\}$ and $\mathfrak{M}=\langle 729,34..36\rangle$ (Corollary 4); $c=74043$ with symbol $\{19\to 9\leftarrow 433\}$ and either $\mathfrak{M}=\langle 2187,65|67\rangle$ with $d_2\left(\mathfrak{M}\right)=5$ or $\mathfrak{M}=\langle 6561,714..719|738..743\rangle$ with $d_2\left(\mathfrak{M}\right)=4$; and $c=82327$ with symbol $\{7\to 19\leftarrow 619\}$ and $\mathfrak{M}=\langle 729,37..39\rangle$ (Corollary 4).

We also have  singular   cases: $c=30457$ with symbol $\{7\to 19\leftarrow 229\}$ and $\mathfrak{M}=\langle 729,37..39\rangle$ (Corollary 4); $c=34029$ with symbol $\{19\to 9\leftarrow 199\}$ and $\mathfrak{M}=\langle 2187,248|249\rangle$; $c=41839$ with symbol $\{43\to 7\leftarrow 139\}$ and $\mathfrak{M}=\langle 6561,693..698\rangle$.

Finally, there is the  super-singular $c=83817$ with symbol $\{9\to 67\leftarrow 139\}$ and $\mathfrak{M}=\langle 6561,693..698\rangle$.

With the exception of $\langle 81,7\rangle$, all groups have non-metabelian descendants and extensions.

In

Table 9. Prototypes for Graph II.1.

No.	c	$\mathit{p}\to \mathit{q}\leftarrow \mathit{r}$	$\mathit{v}1,\mathit{v}4$	${\mathit{y}}_1,{\mathit{y}}_4$	Capitulation Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	3913	$13\to 7\leftarrow 43$	$3,3$	$1,1$	$\mathrm{a}.3^{*}$	${\langle 81,7\rangle }^2$	$=2$
9	$\mathrm{22,581}$	$9\to 193\leftarrow 13$	$3,3$	$0,0$	$\mathrm{d}.19$	${\langle 729,41\rangle }^2$	$\ge 2$
11	$\mathrm{25,929}$	$9\to 67\leftarrow 43$	$3,3$	$0,0$	$\mathrm{b}.10$	${\langle 729,34..36\rangle }^2$	$\ge 2$
15	$\mathrm{30,457}$	$7\to 19\leftarrow 229$	$4,4$	$1,1$	$\mathrm{b}.10$	${\langle 729,37..39\rangle }^2$	$\ge 2$
18	$\mathrm{34,029}$	$19\to 9\leftarrow 199$	$4,4$	$1,0$	$\mathrm{d}.19$	${\langle 2187,248|249\rangle }^2$	$\ge 2$
23	$\mathrm{41,839}$	$43\to 7\leftarrow 139$	$4,4$	$0,0$	$\mathrm{b}.10$	${\langle 6561,693..698\rangle }^2$	$\ge 2$
35	$\mathrm{74,043}$	$19\to 9\leftarrow 433$	$3,3$	$0,0$	$\mathrm{H}.4$	${\langle 2187,65|67\rangle }^2$	$\ge 3$
					or	$\langle 6561,714..719|738..743\rangle$	$\ge 2$
39	$\mathrm{82,327}$	$7\to 19\leftarrow 619$	$3,3$	$0,0$	$\mathrm{b}.10$	${\langle 729,37..39\rangle }^2$	$\ge 2$
42	$\mathrm{83,817}$	$9\to 67\leftarrow 139$	$5,4$	$0,1$	$\mathrm{b}.10$	${\langle 6561,693..698\rangle }^2$	$\ge 2$

, we summarize the prototypes of Graph $\mathrm{II}.1$ in the same manner as in Table 5, except that regularity or (super-)singularity is expressed by 3-valuations $v_{\nu }=v_3(\#\mathrm{Cl}\left(k_{\nu }\right))$ of class numbers of critical fields $k_{\nu }$, $\nu =1,4$, and critical exponents are $y_{\nu }$ in principal factors $A\left(k_{\nu }\right)=p^{x_{\nu }}{q}^{y_{\nu }}{r}^{z_{\nu }}$, $\nu =1,4$.

6.4. Category II, Graph 2

Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 2 of Category $\mathrm{II}$ with combined cubic residue symbol ${[p,q,r]}_3=\{q\leftarrow p\to r\leftarrow q\}$.

Lemma 7

(3-class ranks of components for $\mathrm{II}.2$). Under the normalizing assumption that r splits in $k_{pq}$, precisely the two components $k_1$ and $k_2$ of the quartet have the elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k_1\right)\simeq {\mathrm{Cl}}_3\left(k_2\right)\simeq (3,3)$ of rank 2, whereas the other two components have the 3-class rank ${\varrho }_3\left(k_3\right)={\varrho }_3\left(k_4\right)=3$. Thus, the tame condition $9\mid h_3\left(B_j\right)=(U_j:V_j)\in \{9,27\}$, $r_j=2$, is only satisfied for the bicyclic bicubic fields $B_j$ with $j\in \{1,2,7\}$.

Proof. 

Since $p\to r$, p splits in $k_r$. Since $r\leftarrow q$, q splits in $k_r$, and also splits in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$. By the normalizing assumption that r splits in $k_{pq}$, it also splits in $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$ and $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$. The primes $p,q,r$ share the common decomposition type $(e,f,g)=(3,1,3)$ in the bicyclic bicubic field $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, which implies that ${\varrho }_3\left(k_3\right)={\varrho }_3\left(k_4\right)=3$, according to [2] (Prop. 4.4, pp. 43–44). Finally, only $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$ do not contain $k_3$, $k_4$. □

Proposition 8

(Sub-doublet with 3-rank two for $\mathrm{II}.2$). For fixed $\mu \in \{1,2\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$; then, the principal factor of $k_{\mu }$ is $A\left(k_{\mu }\right)=p$, with $\left[\mathfrak{p}\right]=1$, and the 3-class group of $k_{\mu }$ is

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3).$$

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The unramified cyclic cubic relative extensions of $k_{\mu }$ are among the absolutely bicyclic bicubic fields $B_i$, $1\le i\le 10$. The unique $B_{\mu }$, $\mu \in \{1,2\}$, which only contains $k_{\mu }$, has the norm class group $N_{B_{\mu }/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{\mu }\right)\right)=\langle \left[\mathfrak{r}\right]\rangle$, the transfer kernel

$$ker\left(T_{B_{\mu }/k_{\mu }}\right)\ge \langle \left[\mathfrak{q}\right]\rangle$$

and the 3-class group ${\mathrm{Cl}}_3\left(B_{\mu }\right)=\langle \left[{\mathfrak{R}}_1\right],\left[{\mathfrak{R}}_2\right],\left[{\mathfrak{R}}_3\right]\rangle \ge (3,3)$, generated by the classes of the prime ideals of $B_{\mu }$ over $\mathfrak{r}{\mathcal{O}}_{B_{\mu }}={\mathfrak{R}}_1{\mathfrak{R}}_2{\mathfrak{R}}_3$. The unique $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, which contains $k_1$ and $k_2$, has norm class group $N_{B_7/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_7\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$, the  cyclic   transfer kernel

$$ker\left(T_{B_7/k_{\mu }}\right)=\langle \left[\mathfrak{r}\right]\rangle$$

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of order 3, and  elementary tricyclic  3-class group ${\mathrm{Cl}}_3\left(B_7\right)=\langle \left[{\mathfrak{Q}}_1\right],\left[{\mathfrak{Q}}_2\right],\left[{\mathfrak{Q}}_3\right]\rangle \simeq (3,3,3)$, generated by the classes of the prime ideals of $B_7$ over $\mathfrak{q}{\mathcal{O}}_{B_7}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$. The remaining two $B_i>k_{\mu }$, $i\in \{5,6,8,9\}$, more precisely, $i\in \{5,6\}$ for $\mu =1$, and $i\in \{8,9\}$ for $\mu =2$, have the norm class group $\langle \left[\mathfrak{qr}\right]\rangle$ or $\langle \left[{\mathfrak{qr}}^2\right]\rangle$, and the transfer kernel

$$ker\left(T_{B_i/k_{\mu }}\right)\ge \langle \left[\mathfrak{q}\right]\rangle .$$

In terms of  decisive principal factors $A\left(k_{\nu }\right)=p^{x_{\nu }}{q}^{y_{\nu }}{r}^{z_{\nu }}$ for $\nu \in \{3,4\}$, the ranks of principal factor matrices $M_i$, $i\in \{3,4,5,6,8,9\}$, of  wild   bicyclic bicubic fields are

$$r_3=r_5=r_9=3\mathit{iff}{z}_3\ne 0\mathit{iff}r\mid A\left(k_3\right)\mathit{and}{r}_4=r_6=r_8=3\mathit{iff}{z}_4\ne 0\mathit{iff}r\mid A\left(k_4\right).$$

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Proof. 

Since $q\leftarrow p$, two principal factors are $A\left(k_{pq}\right)=A\left({\tilde{k}}_{pq}\right)=p$; since $p\to r$, two further principal factors are $A\left(k_{pr}\right)=A\left({\tilde{k}}_{pr}\right)=p$; since $r\leftarrow q$, two further principal factors are $A\left(k_{qr}\right)=A\left({\tilde{k}}_{qr}\right)=q$, each by Proposition 3. Since $q\leftarrow p\to r$ is universally repelling, we have $A\left(k_1\right)=A\left(k_2\right)=p$, by Proposition 4.

Thus $\mathfrak{p}=\alpha {\mathcal{O}}_{k_{\mu }}$ is a principal ideal with trivial class $\left[\mathfrak{p}\right]=1$, for $\mu \in \{1,2\}$, whereas the classes $\left[\mathfrak{q}\right],\left[\mathfrak{r}\right]$ are non-trivial. We propose $A\left(k_{\nu }\right)=p^{x_{\nu }}{q}^{y_{\nu }}{r}^{z_{\nu }}$ for $\nu \in \{3,4\}$.

Since the tame condition $9\mid h_3\left(B_j\right)=(U_j:V_j)$ is satisfied for $j\in \{1,2,7\}$, the rank of the corresponding principal factor matrix $M_j$ must be $r_1=r_2=r_7=2$. Due to the principal factors $A\left(k_1\right)=A\left(k_2\right)=p$, this also follows by direct calculation, but has no further consequences. For every wild bicyclic bicubic field $B_j$, $j\in \{3,4,5,6,8,9,10\}$, the rank $r_j$ is calculated with row operations on the associated principal factor matrices $M_j$:

$M_3=\left(\begin{array}{ccc}{x}_3& y_3& z_3\\ 1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$, $M_5=\left(\begin{array}{ccc}1& 0& 0\\ x_3& y_3& z_3\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$, $M_9=\left(\begin{array}{ccc}1& 0& 0\\ x_3& y_3& z_3\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$.

For $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $M_3$ leads to the decisive pivot element $z_3$ in the last column; similarly, for $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $M_5$ leads to $z_3$; and similarly, for $B_9=k_2{k}_3{k}_q{k}_{pr}$, $M_9$ leads to $z_3$. So, $r_3=r_5=r_9=3$ iff $z_3\ne 0$ iff $r\mid A\left(k_3\right)$. Next, we consider:

$M_4=\left(\begin{array}{ccc}{x}_4& y_4& z_4\\ 1& 0& 0\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$, $M_6=\left(\begin{array}{ccc}1& 0& 0\\ x_4& y_4& z_4\\ 0& 1& 0\\ 1& 0& 0\end{array}\right)$, $M_8=\left(\begin{array}{ccc}1& 0& 0\\ x_4& y_4& z_4\\ 1& 0& 0\\ 0& 1& 0\end{array}\right)$.

For $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_4$ leads to the decisive pivot element $z_4$ in the last column; similarly, for $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $M_6$ leads to $z_4$; and similarly, for $B_8=k_2{k}_4{k}_p{k}_{qr}$, $M_8$ leads to $z_4$. So, $r_4=r_6=r_8=3$ iff $z_4\ne 0$ iff $r\mid A\left(k_4\right)$.

By Lemma 2, the minimal subfield unit index $(U_j:V_j)=3$ for $r_j=3$ corresponds to the maximal unit norm index $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=3$, associated to a total transfer kernel $\#ker\left(T_{B_j/k_{\mu }}\right)=9$.

As mentioned in the proof of Lemma 7 already:

Since r splits in $k_{pq}$, it also splits in $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, i.e., $\mathfrak{r}{\mathcal{O}}_{B_{\mu }}={\mathfrak{R}}_1{\mathfrak{R}}_2{\mathfrak{R}}_3$.

Since q splits in $k_r$, it also splits in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, i.e., $\mathfrak{q}{\mathcal{O}}_{B_7}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$.

Since $\mathfrak{q}$ is principal in $k_q$, $k_{qr}$, ${\tilde{k}}_{qr}$, $\left[\mathfrak{q}\right]$ capitulates in $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$; since $\mathfrak{r}$ is principal in $k_r$, $\left[\mathfrak{r}\right]$ capitulates in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$ (Proposition 2). This gives a transposition, either $(1,7)$ or $(2,7)$.

The minimal unit norm index $(U\left(k_{\mu }\right):N_{B_7/k_{\mu }}\left(U_7\right))=1$, associated with the partial transfer kernel $ker\left(T_{B_7/k_{\mu }}\right)=\langle \left[\mathfrak{r}\right]\rangle$, corresponds to the maximal subfield unit index $h_3\left(B_7\right)=(U_7:V_7)=27$, giving rise to the elementary tricyclic-type invariants ${\mathrm{Cl}}_3\left(B_7\right)=\langle \left[{\mathfrak{Q}}_1\right],\left[{\mathfrak{Q}}_2\right],\left[{\mathfrak{Q}}_3\right]\rangle \simeq (3,3,3)$. □

In terms of capitulation targets in Corollary 2, Proposition 8 and parts of its proof are now summarized in

Table 10. Norm class groups and minimal transfer kernels for Graph II.2.

Base	${\mathit{k}}_1$				${\mathit{k}}_2$
Ext	${\mathit{B}}_1$	${\mathit{B}}_5$	${\mathit{B}}_6$	${\mathit{B}}_7$	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$
NCG	$\mathfrak{r}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$
TK	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{q}$
$\varkappa$	$\mathbf{4}$	4	4	$\mathbf{1}$	$\mathbf{2}$	$\mathbf{1}$	2	2

with transposition in bold font.

Theorem 13

(Second 3-class group for $\mathrm{II}.2$). Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 2 of Category $\mathrm{II}$ with combined cubic residue symbol ${[p,q,r]}_3=\{q\leftarrow p\to r\leftarrow q\}$. Without loss of generality, suppose that r splits in $k_{pq}$, and thus, ${\mathrm{Cl}}_3\left(k_1\right)\simeq {\mathrm{Cl}}_3\left(k_2\right)\simeq (3,3)$, and ${\varrho }_3\left(k_3\right)={\varrho }_3\left(k_4\right)=3$.

Then, the  minimal transfer kernel type   (mTKT) of $k_{\mu }$, $1\le \mu \le 2$, is ${\varkappa }_0=\left(2111\right)$, type $\mathrm{H}.4$, and the other possible capitulation types in ascending order ${\varkappa }_0<{\varkappa }^{\prime }<{\varkappa }^{″}<{\varkappa }^{‴}$ are ${\varkappa }^{\prime }=\left(2110\right)$, type $\mathrm{d}.19$, ${\varkappa }^{″}=\left(2100\right)$, type $\mathrm{b}.10$, and ${\varkappa }^{‴}=\left(2000\right)$, type $\mathrm{a}.3^{*}$.

To identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $1\le \mu \le 2$, let the decisive  principal factors   of $k_{\nu }$, $3\le \nu \le 4$, be $A\left(k_{\nu }\right)=p^{x_{\nu }}{q}^{y_{\nu }}{r}^{z_{\nu }}$, and additionally, assume the  regular  situation where both ${\mathrm{Cl}}_3\left(k_3\right)\simeq {\mathrm{Cl}}_3\left(k_4\right)\simeq (3,3,3)$ are elementary tricyclic. Then,

$$\mathfrak{M}\simeq \left\{\begin{array}{cc}\langle 81,7\rangle ,\alpha =[111,11,11,11],\varkappa =\left(2000\right)\hfill & \mathit{if}{z}_3\ne 0,z_4\ne 0,\mathcal{N}=1,\hfill \\ \langle 729,34..39\rangle ,\alpha =[111,111,21,21],\varkappa =\left(2100\right)\hfill & \mathit{if}{z}_3=z_4=0,\mathcal{N}=2,\hfill \\ \langle 729,41\rangle ,\alpha =[111,111,22,21],\varkappa =\left(2110\right)\hfill & \mathit{if}{z}_3=z_4=0,\mathcal{N}=3,\hfill \\ \langle 6561,714..719|738..743\rangle \mathit{or}\hfill & \\ \langle 2187,65|67\rangle ,\alpha =[111,111,22,22],\varkappa =(2111)\hfill & \mathit{if}{z}_3=z_4=0,\mathcal{N}=4,\hfill \end{array}\right.$$

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where $\mathcal{N}:=\#\{1\le j\le 10\mid k_{\mu }<B_j,I_j=27\}$. Only in the leading row has the 3-class field tower warranted the group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$, with length ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$. Otherwise, even if the relation rank $d_2\left(\mathfrak{M}\right)\le 4$ is admissible, the tower length ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$ cannot be excluded.

Proof. 

We give the proof for $k_1$ with unramified cyclic cubic extensions $B_1,B_5,B_6,B_7$ (the proof for $k_2$ with unramified cyclic cubic extensions $B_2,B_7,B_8,B_9$ is similar). We know that the tame ranks are $r_1=r_2=r_7=2$, and thus $I_1,I_2,I_7\in \{9,27\}$, in particular, $I_7=27$, whence certainly $\mathcal{N}\ge 1$. Further, the wild ranks are $r_4=r_6=r_8=3$ iff $z_4\ne 0$, and $r_3=r_5=r_9=3$ iff $z_3\ne 0$.

In the regular situation where the 3-class groups of $k_3$ and $k_4$ are elementary tricyclic, tight bounds arise for the abelian quotient invariants $\alpha$ of the group $\mathfrak{M}$:

The first scenario, $z_3\ne 0$, $z_4\ne 0$, is equivalent to $\mathcal{N}=1$, $h_3\left(B_5\right)=h_3\left(B_9\right)=\frac{1}{3}{h}_3\left(k_3\right)=9$, $h_3\left(B_6\right)=h_3\left(B_8\right)=\frac{1}{3}{h}_3\left(k_4\right)=9$, $h_3\left(B_1\right)=I_1=h_3\left(B_2\right)=I_2=9$, $h_3\left(B_7\right)=I_7=27$, that is, $\alpha =[111,11,11,11]$ and consequently $\varkappa =\left(2000\right)$, since $\langle 81,7\rangle$ is unique with this $\alpha$.

The other three scenarios share $z_3=z_4=0$, and an explicit transposition between $B_1$ and $B_7$, giving rise to $\varkappa =(21\ast \ast )$, and common $h_3\left(B_1\right)=I_1=27$, $\alpha =[111,111,\ast ,\ast ]$.

The second scenario with $\mathcal{N}=2$ is supplemented by $h_3\left(B_5\right)=h_3\left(k_3\right)=27$, $h_3\left(B_6\right)=h_3\left(k_4\right)=27$, giving rise to $\alpha =[111,111,21,21]$, $\varkappa =\left(2100\right)$, characteristic for $\langle 729,34..39\rangle$ (Corollary 4).

The third scenario with $\mathcal{N}=3$ is supplemented by $h_3\left(B_5\right)=3h_3\left(k_3\right)=81$, $h_3\left(B_6\right)=h_3\left(k_4\right)=27$, giving rise to $\alpha =[111,111,22,21]$, $\varkappa =\left(2110\right)$, characteristic for $\langle 729,41\rangle$.

The fourth scenario with $\mathcal{N}=4$ is supplemented by $h_3\left(B_5\right)=3h_3\left(k_3\right)=81$, $h_3\left(B_6\right)=3h_3\left(k_4\right)=81$, giving rise to $\alpha =[111,111,22,22]$, $\varkappa =\left(2111\right)$, characteristic for $\langle 2187,65|67\rangle$ or $\langle 6561,714..719|738..743\rangle$ with coclass $\mathrm{cc}=3$. If $d_2\left(\mathfrak{M}\right)=5$, then ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$. □

Corollary 8

(Uniformity of the sub-doublet for $\mathrm{II}.2$). The components of the sub-doublet with 3-rank two share a common capitulation type $\varkappa \left(k_1\right)\sim \varkappa \left(k_2\right)$, common abelian-type invariants $\alpha \left(k_1\right)\sim \alpha \left(k_2\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_1\right)/k_1)\simeq \mathrm{Gal}({\mathrm{F}}_3^2\left(k_2\right)/k_2)$.

Proof. 

This follows immediately from Theorem 13 and Table 10. □

Example 4.

For 45 examples see [8] (Tbl. 6.16–6.17, pp. 61–62). Prototypes for Graph $\mathrm{II}.2$, i.e., minimal conductors for each scenario in Theorem 13 have been found for each $\mathcal{N}\in \{1,2,3,4\}$.

There are  regular   cases: $c=6327$ with symbol $\{19\to 9\leftarrow 37\to 19\}$ and $\mathfrak{G}=\mathfrak{M}=\langle 81,7\rangle$; $c=41629$ with symbol $\{19\to 313\leftarrow 7\to 19\}$ and $\mathfrak{M}=\langle 729,34..36\rangle$ (Corollary 4); $c=56547$ with symbol $\{61\to 103\leftarrow 9\to 61\}$ and $\mathfrak{M}=\langle 729,41\rangle$; and, with  considerable statistic delay, $c=389329$ with ordinal number 207, symbol $\{19\to 661\leftarrow 31\to 19\}$ and either $\mathfrak{M}=\langle 2187,65|67\rangle$ with $d_2\left(\mathfrak{M}\right)=5$ or $\mathfrak{M}=\langle 6561,714..719|738..743\rangle$ with $d_2\left(\mathfrak{M}\right)=4$.

Further, there are  singular   cases: $c=27873$ with symbol $\{19\to 9\leftarrow 163\to 19\}$ and $\mathfrak{M}=\langle 729,34..36\rangle$ (Corollary 4); $c=29197$ with symbol $\{43\to 7\leftarrow 97\to 43\}$ and $\mathfrak{M}=\langle 2187,253\rangle$; and $c=63511$ with symbol $\{43\to 7\leftarrow 211\to 43\}$ and $\mathfrak{M}=\langle 729,37..39\rangle$ (Corollary 4).

Finally, there is the  super-singular $c=66157$ with symbol $\{13\to 7\leftarrow 727\to 13\}$ and $\mathfrak{M}=\langle 6561,1989\rangle$.

With exception of $\langle 81,7\rangle$, all groups have non-metabelian descendants and extensions.

In

Table 11. Prototypes for Graph II.2.

No.	c	$\mathit{q}\to \mathit{r}\leftarrow \mathit{p}\to \mathit{q}$	$\mathit{v}3,\mathit{v}4$	${\mathit{z}}_3,{\mathit{z}}_4$	Capitulation Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	6327	$19\to 9\leftarrow 37\to 19$	$3,3$	$1,1$	$\mathrm{a}.3^{*}$	${\langle 81,7\rangle }^2$	$=2$
8	$\mathrm{27,873}$	$19\to 9\leftarrow 163\to 19$	$4,4$	$1,1$	$\mathrm{b}.10$	${\langle 729,34..36\rangle }^2$	$\ge 2$
10	$\mathrm{29,197}$	$43\to 7\leftarrow 97\to 43$	$4,4$	$0,1$	$\mathrm{b}.10$	${\langle 2187,253\rangle }^2$	$\ge 3$
14	$\mathrm{41,629}$	$19\to 313\leftarrow 7\to 19$	$3,3$	$0,0$	$\mathrm{b}.10$	${\langle 729,34..36\rangle }^2$	$\ge 2$
23	$\mathrm{56,547}$	$61\to 103\leftarrow 9\to 61$	$3,3$	$0,0$	$\mathrm{d}.19$	${\langle 729,41\rangle }^2$	$\ge 2$
28	$\mathrm{63,511}$	$43\to 7\leftarrow 211\to 43$	$4,4$	$1,1$	$\mathrm{b}.10$	${\langle 729,37..39\rangle }^2$	$\ge 2$
31	$\mathrm{66,157}$	$13\to 7\leftarrow 727\to 13$	$5,4$	$1,0$	$\mathrm{d}.19$	${\langle 6561,1989\rangle }^2$	$\ge 2$
207	$\mathrm{389,329}$	$19\to 661\leftarrow 31\to 19$	$3,3$	$0,0$	$\mathrm{H}.4$	${\langle 2187,65|67\rangle }^2$	$\ge 3$
					or	$\langle 6561,714..719|738..743\rangle$	$\ge 2$

, we summarize the prototypes of Graph $\mathrm{II}.2$ in the same way as in Table 5, except that regularity or (super-)singularity is expressed by 3-valuations $v_{\nu }=v_3(\#\mathrm{Cl}\left(k_{\nu }\right))$ of class numbers of critical fields $k_{\nu }$, $\nu =3,4$, and critical exponents are $z_{\nu }$ in principal factors $A\left(k_{\nu }\right)=p^{x_{\nu }}{q}^{y_{\nu }}{r}^{z_{\nu }}$, $\nu =3,4$.

7. Category III, Graphs 1–4

Let the combined cubic residue symbol ${[p,q,r]}_3$ of three prime(power)s dividing the conductor $c=pqr$ be either $\{p,q,r;\delta \not\equiv 0(\mathrm{mod}3\left)\right\}$ or $\{p\to q;r\}$ or $\{p\to q\to r\}$ or $\{p\to q\to r\to p\}$. The symbol does not contain any mutual cubic residues. We verify a conjecture in [8] (Cnj. 1, p. 48).

Theorem 14.

A cyclic cubic field k with conductor $c=pqr$, divisible by exactly three prime(power)s $p,q,r$, has an abelian 3-class field tower with group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k\right)/k)\simeq \langle 9,2\rangle$, $\alpha =[1,1,1,1]$, $\varkappa =\left(0000\right)$, if and only if the primes $p,q,r$ form one of the four Graphs 1–4 of Category $\mathrm{III}$.

Proof. 

Ayadi [2] (Thm. 4.1, pp. 76–77) has proved the sufficiency of the condition. He does not claim explicitly that the condition is also necessary. However, his techniques are able to prove both directions. Recall that for both Graphs 1–2 of Categories $\mathrm{I}$ and $\mathrm{II}$, there is at least one component of the quartet ${\left(k_{\mu }\right)}_{\mu =1}^4$ with 3-class rank $\varrho \left(k_{\mu }\right)=3$, and that for all Graphs 5–9 of Category $\mathrm{III}$, two primes $p\leftrightarrow q$ are mutual cubic residues, according to Theorem 2. In contrast, precisely for Graphs 1–4 of Category $\mathrm{III}$, the symbol ${[p,q,r]}_3$ does not contain any mutual cubic residues, and all four components have the 3-class rank $\varrho \left(k_{\mu }\right)=2$ and the elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$, whence these are the only cases where all bicyclic bicubic fields $B_j$, $1\le j\le 10$, satisfy the tame relation $h_3\left(B_j\right)=(U_j:V_j)=3$ with the matrix rank $r_j=3$. This is equivalent with abelian-type invariants $\alpha \left(k_{\mu }\right)=[1,1,1,1]$ for all $1\le \mu \le 4$. By the strategy of pattern recognition [15], this enforces the abelian group $\mathfrak{G}\simeq \langle 9,2\rangle \simeq (3,3)$, which is the unique 3-group G with $G/G^{\prime }\simeq (3,3)$ and abelian-quotient invariants $\alpha \left(G\right)=[1,1,1,1]$. □

For the prototypes of Graphs $1,\dots ,4$ of Category $\mathrm{III}$ see [8] (Tbl. 6.3, p. 48). Systematic tables are presented at http://www.algebra.at/ResearchFrontier2013ThreeByThree.htm (accessed on 10 October 2023) in Section 2.1 and Section 2.2.

8. Category III, Graphs 5–9

In this section, the combined cubic residue symbol ${[p,q,r]}_3$ of three prime(power)s dividing the conductor $c=pqr$ contains a unique pair $p\leftrightarrow q$ of mutual cubic residues.

Consequently, the decisive principal factors

$$A\left(k_{pq}\right)=p^m{q}^n,A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$$

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must be assumed with variable exponents in $\{0,1,2\}$, such that $(m,n)\ne (0,0)$ and $(\tilde{m},\tilde{n})\ne (0,0)$. Concerning 3-class groups of cyclic cubic subfields $k<k^{*}$ with $t=2$, an elementary cyclic group ${\mathrm{Cl}}_3\left(k\right)\simeq \left(3\right)$ is warranted for $k\in \{k_{pr},{\tilde{k}}_{pr},k_{qr},{\tilde{k}}_{qr}\}$. For the critical fields $k\in \{k_{pq},{\tilde{k}}_{pq}\}$, however, we must distinguish the regular situation ${\mathrm{Cl}}_3\left(k_f^{*}\right)\simeq (3,3)$ in terms of the sub-genus field $k_f^{*}=k_{pq}·{\tilde{k}}_{pq}$ with partial conductor $f=pq$, which divides $c=pqr$, where ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq {\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)\simeq (3,3)$ and equality $(m,n)=(\tilde{m},\tilde{n})$ is warranted, as opposed to the singular situation ${\mathrm{Cl}}_3\left(k_f^{*}\right)\simeq (3,3,3)$ and the super-singular situation $81\mid h_3\left(k_f^{*}\right)$, where usually, ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq {\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)\simeq (9,3)$.

For doublets $(k_{pq},{\tilde{k}}_{pq})$ with conductor $f=pq$ and a non-elementary bicyclic 3-class group, a distinction arises from the 3-valuation $v^{*}:=v_3\left(h\left(k_f^{*}\right)\right)$ of the class number of the 3-genus field $k_f^{*}$:

Definition 6.

A quartet ${\left(k_{\mu }\right)}_{1\le \mu \le 4}$ with conductor $c=pqr$ and its sub-doublet $(k_{pq},{\tilde{k}}_{pq})$ of cyclic cubic fields with common partial conductor $f=pq$ is called

$$\left\{\begin{array}{cc}\mathbf{regular}\hfill & \mathit{if}{v}^{*}\in \{0,1,2\},\hfill \\ \mathbf{singular}\hfill & \mathit{if}{v}^{*}=3,\hfill \\ \mathbf{super}-\mathbf{singular}\hfill & \mathit{if}{v}^{*}\in \{4,5,6,\dots \}.\hfill \end{array}\right.$$

(49)

Let $(k_1,\dots ,k_4)$ be the quartet of cyclic cubic number fields sharing the common discriminant $d=c^2$ with conductor $c=pqr$, divisible by exactly three primes $\equiv 1\left(\mathrm{mod}3\right)$ (one among them may be the prime power $3^2$), and belonging to one of the Graphs 5–9 of Category $\mathrm{III}$. According to Theorem 2, ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$ and thus $h_3\left(k_{\mu }\right)=9$, for $1\le \mu \le 4$.

Due to these facts, the class number relation $243·h_3\left(B_j\right)=(U_j:V_j)·9·9·1·3$ for $j\in \{5,6,8,9\}$ implies that there are precisely four tame bicyclic bicubic fields, $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$, satisfying $9\mid h_3\left(B_j\right)=(U_j:V_j)$, for each $j\in \{5,6,8,9\}$, and so we must have the matrix ranks $r_5=r_6=r_8=r_9=2$ with indices $(U_j:V_j)\in \{9,27\}$.

In contrast, each of the six wild bicyclic bicubic fields, $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, with $h_3\left(B_j\right)>(U_j:V_j)$, either contains $k_{pq}$ or ${\tilde{k}}_{pq}$. The class number relation (16) implies

$$243·h_3\left(B_j\right)=(U_j:V_j)·\left\{\begin{array}{cc}9·h_3\left(k_{pq}\right)·3·3\hfill & \mathrm{for}j=1,2,\hfill \\ 9·9·1·h_3\left(k_{pq}\right)\hfill & \mathrm{for}j=10,\hfill \\ 9·h_3\left({\tilde{k}}_{pq}\right)·3·3\hfill & \mathrm{for}j=3,4,\hfill \\ 9·9·1·h_3\left({\tilde{k}}_{pq}\right)\hfill & \mathrm{for}j=7.\hfill \end{array}\right.$$

Summarized, in dependence on the index $I_j:=(U_j:V_j)$ of subfield units and the rank $r_j$,

$$h_3\left(B_j\right)=\left\{\begin{array}{cc}{h}_3\left(k_{pq}\right)\hfill & \mathrm{for}j=1,2,10,I_j=3,r_j=3,\hfill \\ 3·h_3\left(k_{pq}\right)\hfill & \mathrm{for}j=1,2,10,I_j=9,r_j=2,\hfill \\ 9·h_3\left(k_{pq}\right)\hfill & \mathrm{for}j=1,2,10,I_j=27,r_j=2,\hfill \\ h_3\left({\tilde{k}}_{pq}\right)\hfill & \mathrm{for}j=3,4,7,I_j=3,r_j=3,\hfill \\ 3·h_3\left({\tilde{k}}_{pq}\right)\hfill & \mathrm{for}j=3,4,7,I_j=9,r_j=2,\hfill \\ 9·h_3\left({\tilde{k}}_{pq}\right)\hfill & \mathrm{for}j=3,4,7,I_j=27,r_j=2,\hfill \end{array}\right.$$

(50)

with $h_3\left(k_{pq}\right)=h_3\left({\tilde{k}}_{pq}\right)=9$ in the regular situation, and $h_3\left(k_{pq}\right),h_3\left({\tilde{k}}_{pq}\right)\ge 27$ in the singular or super-singular situation. Formula (50) supplements Corollary 1 in the case $p\leftrightarrow q$.

Lemma 8

(3-class ranks of components). All four components $k_{\mu }$, $1\le \mu \le 4$, of the quartet have elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$. The condition $9\mid h_3\left(B_j\right)=(U_j:V_j)\in \{9,27\}$, $r_j=2$, is satisfied for $j\in \{5,6,8,9\}$, the so-called tame extensions.

Proof. 

This is a consequence of the definition of Graph 5–9 in Category III and the rank distribution in Theorem 2. The fields $B_j$ with $j\in \{5,6,8,9\}$ neither contain $k_{pq}$ nor ${\tilde{k}}_{pq}$. □

All computations for examples in the following subsections were performed with Magma [6,21,22].

8.1. Category III, Graph 5

In this section, the combined cubic residue symbol of three prime(power)s dividing the conductor $c=pqr$ is assumed to be ${[p,q,r]}_3=\{p\leftrightarrow q;r\}$.

Since there are no trivial cubic residue symbols among the three prime(power) divisors $p,q,r$ of the conductor $c=pqr$, except $p\leftrightarrow q$ with overall assumption (48), the principal factors of the subfields $k\in \{k_{pr},{\tilde{k}}_{pr},k_{qr},{\tilde{k}}_{qr}\}$ with $t=2$ of the absolute genus field $k^{*}$ must be divisible by both relevant primes, and we can use the general approach

$$\begin{array}{cc}\hfill A\left(k_{pr}\right)=p^{\mathsf{\ell }}r,& A\left({\tilde{k}}_{pr}\right)=p^{-\mathsf{\ell }}r,\mathrm{and}\hfill \\ \hfill A\left(k_{qr}\right)=q^{s}r,& A\left({\tilde{k}}_{qr}\right)=q^{-s}r,\hfill \end{array}$$

(51)

with $\mathsf{\ell },s\in \{-1,1\}$, identifying $-1\equiv 2\left(\mathrm{mod}3\right)$, since it is easier to manage: ${\mathsf{\ell }}^2=s^2=1$.

Lemma 9.

In dependence on the  decisive principal factors   in Formula (51), the  principal factors   of the quartet ${\left(k_{\mu }\right)}_{\mu =1}^4$ sharing the common conductor $c=pqr$ with Graph $\mathrm{III}.5$ are given by

$$\begin{array}{cc}\hfill A\left(k_1\right)=pq{r}^2,A\left(k_2\right)=pqr,A\left(k_3\right)=p{q}^{2}r,A\left(k_4\right)=p^{2}qr& \mathit{if}(\mathsf{\ell },s)=(1,1),\hfill \\ \hfill A\left(k_1\right)=p^{2}qr,A\left(k_2\right)=p{q}^{2}r,A\left(k_3\right)=pqr,A\left(k_4\right)=pq{r}^2& \mathit{if}(\mathsf{\ell },s)=(1,2),\hfill \\ \hfill A\left(k_1\right)=p{q}^{2}r,A\left(k_2\right)=p^{2}qr,A\left(k_3\right)=pq{r}^2,A\left(k_4\right)=pqr& \mathit{if}(\mathsf{\ell },s)=(2,1),\hfill \\ \hfill A\left(k_1\right)=pqr,A\left(k_2\right)=pq{r}^2,A\left(k_3\right)=p^{2}qr,A\left(k_4\right)=p{q}^{2}r& \mathit{if}(\mathsf{\ell },s)=(2,2),\hfill \\ \hfill A\left(k_1\right)=p^{\mathsf{\ell }}{q}^s{r}^{-1},A\left(k_2\right)=p^{\mathsf{\ell }}{q}^{s}r,A\left(k_3\right)=p^{\mathsf{\ell }}{q}^{-s}r,A\left(k_4\right)=p^{-\mathsf{\ell }}{q}^{s}r& \mathit{generally}.\hfill \end{array}$$

(52)

Proof. 

We implement the general approach (51). From the ranks $r_j=2$ for $j=5,6,8,9$, there arise constraints for the exponents in the proposal $A\left(k_{\mu }\right)=p^{x_{\mu }}{q}^{y_{\mu }}{r}^{z_{\mu }}$, $1\le \mu \le 4$, with the aid of principal factor matrices. For these tame bicyclic bicubic fields $B_j$, $j\in \{5,6,8,9\}$, the rank $r_j$ is calculated with row operations on the associated matrix $M_j$:

$M_5=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_3& y_3& z_3\\ 1& 0& 0\\ 0& -s& 1\end{array}\right)$, $M_6=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_4& y_4& z_4\\ 0& 1& 0\\ -\mathsf{\ell }& 0& 1\end{array}\right)$, $M_8=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ x_4& y_4& z_4\\ 1& 0& 0\\ 0& s& 1\end{array}\right)$, $M_9=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ x_3& y_3& z_3\\ 0& 1& 0\\ \mathsf{\ell }& 0& 1\end{array}\right)$.

For $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $M_5$ leads to the decisive pivot elements $z_1+s{y}_1$ and $z_3+s{y}_3$ in the last column; similarly, for $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $M_6$ leads to $z_1+\mathsf{\ell }{x}_1$ and $z_4+\mathsf{\ell }{x}_4$; similarly, for $B_8=k_2{k}_4{k}_p{k}_{qr}$, $M_8$ leads to $z_2-s{y}_2$ and $z_4-s{y}_4$; and similarly, for $B_9=k_2{k}_3{k}_q{k}_{pr}$, $M_9$ leads to $z_2-\mathsf{\ell }{x}_2$ and $z_3-\mathsf{\ell }{x}_3$. So, $r_5=r_6=r_8=r_9=2$ implies $\mathsf{\ell }{x}_1\equiv s{y}_1\equiv -z_1$, $\mathsf{\ell }{x}_2\equiv s{y}_2\equiv z_2$, $\mathsf{\ell }{x}_3\equiv -s{y}_3\equiv z_3$, $-\mathsf{\ell }{x}_4\equiv s{y}_4\equiv z_4$, and consequently (52). □

Proposition 9

(Quartet with 3-rank two for $\mathrm{III}.5$). Let ${\left(k_{\mu }\right)}_{\mu =1}^4$ be a quartet with common conductor $c=pqr$, whose combined cubic residue symbol belongs to Graph 5 of Category $\mathrm{III}$. Then, the ranks of principal factor matrices of tame bicyclic bicubic fields are $r_j=2$ for $j=5,6,8,9$. In terms of exponents of primes in four variable principal factors, $A\left(k_{pq}\right)=p^m{q}^n$, $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, from (48), and $A\left(k_{pr}\right)=p^{\mathsf{\ell }}r$, $A\left(k_{qr}\right)=q^{s}r$, from (51), the ranks of principal factor matrices of wild bicyclic bicubic fields are given by

$$r_1=r_2=r_{10}=3\mathit{iff}\mathsf{\ell }m\not\equiv -sn\left(\mathrm{mod}3\right)\mathit{and}{r}_3=r_4=r_7=3\mathit{iff}\mathsf{\ell }\tilde{m}\not\equiv s\tilde{n}\left(\mathrm{mod}3\right).$$

(53)

Proof. 

Up to this point, the parameters $m,n,\tilde{m},\tilde{n}$ have not come into play yet. They decide about the rank $r_j$ of the associated principal factor matrices $M_j$ of the wild bicyclic bicubic fields $B_j$, $j\in \{1,2,3,4,7,10\}$. Hence, we perform row operations on these matrices:

$M_1=\left(\begin{array}{ccc}\mathsf{\ell }& s& -1\\ m& n& 0\\ \mathsf{\ell }& 0& 1\\ 0& s& 1\end{array}\right)$, $M_2=\left(\begin{array}{ccc}\mathsf{\ell }& s& 1\\ m& n& 0\\ -\mathsf{\ell }& 0& 1\\ 0& -s& 1\end{array}\right)$, $M_{10}=\left(\begin{array}{ccc}\mathsf{\ell }& -s& 1\\ -\mathsf{\ell }& s& 1\\ 0& 0& 1\\ m& n& 0\end{array}\right)$.

For $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $M_1$ leads to the decisive pivot element $-\mathsf{\ell }m-sn$ in the last column; similarly, for $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $M_2$ leads to $\mathsf{\ell }m+sn$; and similarly, for $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $M_{10}$ leads to $n+\mathsf{\ell }sm$ in the middle column. So, $r_1=r_2=r_{10}=3$ iff $\mathsf{\ell }m\not\equiv -sn$, by viewing the pivot elements modulo 3. Next, we consider:

$M_3=\left(\begin{array}{ccc}\mathsf{\ell }& -s& 1\\ \tilde{m}& \tilde{n}& 0\\ -\mathsf{\ell }& 0& 1\\ 0& s& 2\end{array}\right)$, $M_4=\left(\begin{array}{ccc}-\mathsf{\ell }& s& 1\\ \tilde{m}& \tilde{n}& 0\\ \mathsf{\ell }& 0& 1\\ 0& -s& 1\end{array}\right)$, $M_7=\left(\begin{array}{ccc}\mathsf{\ell }& s& -1\\ \mathsf{\ell }& s& 1\\ 0& 0& 1\\ \tilde{m}& \tilde{n}& 0\end{array}\right)$.

For $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $M_3$ leads to $\mathsf{\ell }\tilde{m}-s\tilde{n}$; similarly, for $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_4$ leads to $-\mathsf{\ell }\tilde{m}+s\tilde{n}$; and similarly, for $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ leads to $\tilde{n}-\mathsf{\ell }s\tilde{m}$ in the middle column. So, $r_3=r_4=r_7=3$ iff $\mathsf{\ell }\tilde{m}\not\equiv s\tilde{n}$. □

In Ayadi’s thesis [2] (p. 80), only the special case $\mathsf{\ell }=s=-1$ is elaborated. As mentioned above already, the condition $(m,n)=(\tilde{m},\tilde{n})$ is warranted in the regular situation $9\parallel h\left(k_{pq}\right)$. In any situation, at least one of the following two rank equations, which imply a total transfer kernel, is satisfied—in many cases, even both simultaneously:

$$\begin{array}{cc}\hfill r_1=r_2=r_{10}=3\mathrm{for}& (m,n)\in \left\{\right(0,1),(1,0\left)\right\},\hfill \\ \hfill r_3=r_4=r_7=3\mathrm{for}& (\tilde{m},\tilde{n})\in \{(0,1),(1,0)\}.\hfill \end{array}$$

(54)

Theorem 15

(Second 3-class groups for $\mathrm{III}.5$). There are several minimal transfer kernel types (mTKT) ${\varkappa }_0$ of $k_{\mu }$, $1\le \mu \le 4$, and other possible capitulation types in ascending order ${\varkappa }_0<\varkappa <{\varkappa }^{\prime }<{\varkappa }^{″}<{\varkappa }^{‴}$, either ${\varkappa }_0=\left(2134\right)$, type $\mathrm{G}.16$; $\varkappa =\left(2130\right)$, type $\mathrm{d}.23$, or ${\varkappa }_0=\left(2143\right)$, type $\mathrm{G}.19$; $\varkappa =\left(2140\right)$, type $\mathrm{d}.25$, ending in ${\varkappa }^{\prime }=\left(2100\right)$, type $\mathrm{b}.10$; ${\varkappa }^{″}=\left(2000\right)$, type $\mathrm{a}.3^{*}$ or $\mathrm{a}.3$, or ${\varkappa }^{″}=\left(0004\right)$, type $\mathrm{a}.2$; and the maximal ${\varkappa }^{‴}=\left(0000\right)$, type $\mathrm{a}.1$.

In terms of the counter ${\mathcal{N}}^{*}:=\#\{1\le j\le 10\mid (U_j:V_j)=27\}$, of  maximal indices of subfield units $I_j=(U_j:V_j)$ for all ten bicyclic bicubic fields $B_j<k^{*}$ with conductor $c=pqr$, the second 3-class groups $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$ are given in the following way as uniform or non-uniform quartets, with the abbreviation $P_7:=\langle 2187,64\rangle$:

$$\mathfrak{M}=\left\{\begin{array}{cc}{\langle 243,28..30\rangle }^4,\alpha =[21,11,11,11],\varkappa =\left(0000\right)\hfill & \mathit{if}{\mathcal{N}}^{*}=0,\hfill \\ \langle 243,27\rangle ,\alpha =[11,11,11,22],\varkappa =\left(0004\right)\hfill & \mathit{once}\mathit{if}{\mathcal{N}}^{*}=1,\hfill \\ {\langle 243,28..30\rangle }^3,\alpha =[21,11,11,11],\varkappa =\left(0000\right)\hfill & \mathit{thrice}\mathit{if}{\mathcal{N}}^{*}=1,\hfill \\ {\langle 81,7\rangle }^4,\alpha =[111,11,11,11],\varkappa =\left(2000\right)\hfill & \mathit{if}{\mathcal{N}}^{*}=2,\hfill \\ {\langle 243,27\rangle }^2,\alpha =[11,11,11,22],\varkappa =\left(0004\right)\hfill & \mathit{twice}\mathit{if}{\mathcal{N}}^{*}=3,\hfill \\ {\langle 243,25\rangle }^2,\alpha =[22,11,11,11],\varkappa =\left(2000\right)\hfill & \mathit{twice}\mathit{if}{\mathcal{N}}^{*}=3,\hfill \\ {\langle 729,34..39\rangle }^4,\alpha =[111,111,21,21],\varkappa =\left(2100\right)\hfill & \mathit{if}{\mathcal{N}}^{*}=4,\hfill \\ {\langle 2187,250\rangle }^2,\alpha =[111,111,32,21],\varkappa =\left(2130\right)\hfill & \mathit{twice}\mathit{if}{\mathcal{N}}^{*}=7,\hfill \\ {\langle 2187,251|252\rangle }^2,\alpha =[111,111,32,21],\varkappa =\left(2140\right)\hfill & \mathit{twice}\mathit{if}{\mathcal{N}}^{*}=7,\hfill \\ {(P_7-\#2;40|48)}^2,\alpha =[111,111,32,32],\varkappa =\left(2134\right)\hfill & \mathit{twice}\mathit{if}{\mathcal{N}}^{*}=10,\hfill \\ (P_7-{\#2;42\left|45\right|49)}^2,\alpha =[111,111,32,32],\varkappa =\left(2143\right)\hfill & \mathit{twice}\mathit{if}{\mathcal{N}}^{*}=10.\hfill \end{array}\right.$$

(55)

The leading six rows concern the  regular  situation ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq (3,3)$. In particular, the condition $(m,n)\in \left\{\right(0,1),(1,0\left)\right\}$ for ${\langle 81,7\rangle }^4$ is equivalent to the extra special group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{pq}\right)/k_{pq})\simeq \langle 27,4\rangle$, whereas $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{pq}\right)/k_{pq})\simeq \langle 9,2\rangle$ is abelian for all other pairs $(m,n)$. The trailing rows concern the  (super-)singular   situation with $h_3\left(k_{pq}\right)=h_3\left({\tilde{k}}_{pq}\right)=27$. With exception of the trailing rows, the 3-class field tower has length ${\mathsf{\ell }}_3\left(K_{\mu }\right)=2$ and group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$.

Proof. 

Let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$.

Since p splits in $k_q$, it also splits in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$.

Since q splits in $k_p$, it also splits in $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$ and $B_8=k_2{k}_4{k}_p{k}_{qr}$. By Corollary 3,

since $\mathfrak{q}$ is principal ideal in $k_q$, the class $\left[\mathfrak{q}\right]$ capitulates in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$;

since $\mathfrak{r}$ is principal ideal in $k_r$, the class $\left[\mathfrak{r}\right]$ capitulates in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$ and $B_{10}=k_3{k}_4{k}_r{k}_{pq}$.

Since $\mathfrak{qr}$ is principal ideal in ${\tilde{k}}_{qr}$, the class $\left[\mathfrak{qr}\right]$ capitulates in $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, and $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, by Proposition 2. Since $A\left(k_1\right)=pqr$ and $A\left(k_3\right)=p^{2}qr$, $\left[\mathfrak{qr}\right]$ generates the same subgroup as $\left[\mathfrak{p}\right]$ in $ker\left(T_{B_5/k_{\mu }}\right)$, $\mu =1,3$.

Since ${\mathfrak{qr}}^2$ is a principal ideal in $k_{qr}$, the class $\left[{\mathfrak{qr}}^2\right]$ capitulates in $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, and $B_8=k_2{k}_4{k}_p{k}_{qr}$, by Proposition 2. Since $A\left(k_2\right)=pq{r}^2$ and $A\left(k_4\right)=p{q}^{2}r$, $\left[{\mathfrak{qr}}^2\right]$ generates the same subgroup as $\left[\mathfrak{p}\right]$ in $ker\left(T_{B_8/k_{\mu }}\right)$, $\mu =2,4$.

The 3-class group of $k_{\mu }$ is always ${\mathrm{Cl}}_3=\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle$. It contains the norm class groups of $B_j>k_{\mu }$ as subgroups of index 3: $N_{B_j/k_{\mu }}{\mathrm{Cl}}_3\left(B_j\right)$ is always generated by $\left[\mathfrak{q}\right]$ for $j=5,8$, due to the above-mentioned splitting of q. See also

Table 12. Norm class groups and minimal transfer kernels for Graph $\mathrm{III}.5$.

Base	${\mathit{k}}_1$				${\mathit{k}}_2$				${\mathit{k}}_3$				${\mathit{k}}_4$
Ext	${\mathit{B}}_1$	${\mathit{B}}_5$	${\mathit{B}}_6$	${\mathit{B}}_7$	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$	${\mathit{B}}_4$	${\mathit{B}}_6$	${\mathit{B}}_8$	${\mathit{B}}_{10}$
NCG	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{qr}$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{r}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{qr}$
TK	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{r}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$
$\varkappa$	1	$\mathbf{3}$	$\mathbf{2}$	4	1	2	$\mathbf{4}$	$\mathbf{3}$	$\mathbf{4}$	$\mathbf{3}$	$\mathbf{2}$	$\mathbf{1}$	$\mathbf{4}$	$\mathbf{3}$	$\mathbf{2}$	$\mathbf{1}$

.

We recall that equality $(m,n)=(\tilde{m},\tilde{n})$ is warranted for the regular situation ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq (3,3)$, and there is an equivalence involving the counter $\mathcal{P}$ in Theorem 4: $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{pq}\right)/k_{pq})\simeq \langle 27,4\rangle$ iff $\mathcal{P}=1$ iff (either $m=0$ or $n=0$) iff $(m,n)\in \left\{\right(0,1),(1,0\left)\right\}$. Let $I_j:=(U_j:V_j)$.

${\mathcal{N}}^{*}=0$ implies wild ranks $r_j=2$ and $I_j=9$, $h_3\left(B_j\right)=3·h_3\left(k_{pq}\right)=3·9=27$ for $j\in \{1,2,10\}$, but $r_j=3$, $I_j=3$, $h_3\left(B_j\right)=h_3\left({\tilde{k}}_{pq}\right)=9$ for $j\in \{3,4,7\}$, according to Formula (50), and tame indices $I_j=9$ for all $j\in \{5,6,8,9\}$. The uniform minimal indices $I_j$ of subgroup units correspond to maximal norm unit indices $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}U\left(B_j\right))=3$ and thus to total capitulations whenever $k_{\mu }<B_j$ is a subfield for $1\le j\le 10$, $1\le \mu \le 4$. According to Theorem 9 and Corollary 4, the resulting abelian-type invariants $\alpha =[21,11,11,11]$ and transfer kernel type $\varkappa =\left(0000\right)$, that, is the Artin pattern $(\alpha ,\varkappa )$, identify three possible groups, $\mathfrak{M}\simeq \langle 243,28..30\rangle$, since $\langle 81,9\rangle$ must be cancelled due to the wrong second layer ${\alpha }_2$.

An exception arises for ${\mathcal{N}}^{*}=1$, which causes non-uniformity with $I_1=27$, $h_3\left(B_1\right)=9·h_3\left(k_{pq}\right)=9·9=81$, as opposed to the remaining $I_2=I_{10}=9$. (Everything else is like ${\mathcal{N}}^{*}=0$.) Thus, $(U\left(k_1\right):N_{B_1/k_1}U\left(B_1\right))=1$, and here, we have a fixed-point capitulation, $ker\left(T_{B_1/k_1}\right)=\langle \left[{\mathfrak{qr}}^2\right]\rangle$. The corresponding abelian-type invariants $\alpha =[11,11,11,22]$ and transfer kernel type $\varkappa =\left(0004\right)$ uniquely identify the group $\langle 243,27\rangle$ for $\mu =1$. The remaining three groups are ${\langle 243,28..30\rangle }^3$.

For ${\mathcal{N}}^{*}=2$ and $(m,n)\in \left\{\right(0,1),(1,0\left)\right\}$, the relations $m\not\equiv -n$ and $m\not\equiv n$ imply $r_j=3$, $I_j=3$, $h_3\left(B_j\right)=h_3\left({\tilde{k}}_{pq}\right)=9$ for all wild bicyclic bicubic fields $B_j$, $j\in \{1,2,3,4,7,10\}$. For $j\in \{5,8\}$, we have $I_j=9$, but for $j\in \{6,9\}$, the maximal index $I_j=27$ is attained and enables an elementary tricyclic 3-class group ${\mathrm{Cl}}_3\left(B_j\right)=\langle {\mathfrak{P}}_1,{\mathfrak{P}}_2,{\mathfrak{P}}_3\rangle \simeq (3,3,3)$ generated by the prime ideals lying over $\mathfrak{p}{\mathcal{O}}_{B_j}={\mathfrak{P}}_1·{\mathfrak{P}}_2·{\mathfrak{P}}_3$. Here, we have a non-fixed-point capitulation $ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$. The transposition is hidden by total capitulation in $B_5$ and $B_8$ with the norm class group generated by $\left[\mathfrak{q}\right]$. The abelian-type invariants $\alpha =[111,11,11,11]$ and the transfer kernel type $\varkappa =\left(2000\right)$ uniquely identify the group $\langle 81,7\rangle \simeq {\mathrm{Syl}}_3\left(A_9\right)$ for $\mu =1$.

${\mathcal{N}}^{*}=3$ implies $r_j=3$ and thus wild indices $I_j=3$, $h_3\left(B_j\right)=h_3\left({\tilde{k}}_{pq}\right)=9$ for $j\in \{1,2,10\}$. We also have tame indices $I_j=9$ for $j\in \{5,6,8,9\}$. However, we have $r_j=2$ and the remaining wild indices $I_j=27$, $h_3\left(B_j\right)=9·h_3\left(k_{pq}\right)=9·9=81$ for $j\in \{3,4,7\}$, according to Formula (50). There arises a fixed-point capitulation, $ker\left(T_{B_7/k_{\mu }}\right)=\langle \left[\mathfrak{r}\right]\rangle$ and, non-uniformly, a non-fixed-point capitulation, $ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{p}\right]\rangle$ for $j=3,4$ with norm class groups also generated by $\left[\mathfrak{r}\right]$. The corresponding abelian-type invariants $\alpha =[11,11,11,22]$ and transfer kernel types $\varkappa =\left(0004\right)$ and $\varkappa =\left(0003\right)$, respectively, uniquely identify the two groups ${\langle 243,27\rangle }^2$ and the remaining two groups ${\langle 243,25\rangle }^2$, respectively.

For ${\mathcal{N}}^{*}=4$ and the simplest singular or super-singular situation with $h_3\left(k_{pq}\right)=h_3\left({\tilde{k}}_{pq}\right)=27$, $\mathcal{P}=1$ implies $r_j=3$, $I_j=3$, $h_3\left(B_j\right)=27$ for all wild $j\in \{1,2,3,4,7,10\}$, and uniformly, $h_3\left(B_j\right)=I_j=27$ for all tame $j\in \{5,6,8,9\}$. The latter correspond to the elementary tricyclic 3-class groups ${\mathrm{Cl}}_3\left(B_j\right)=\langle {\mathfrak{Q}}_1,{\mathfrak{Q}}_2,{\mathfrak{Q}}_3\rangle \simeq (3,3,3)$ generated by the prime ideals lying over $\mathfrak{q}{\mathcal{O}}_{B_j}={\mathfrak{Q}}_1·{\mathfrak{Q}}_2·{\mathfrak{Q}}_3$ for $j=5,8$, and ${\mathrm{Cl}}_3\left(B_j\right)=\langle {\mathfrak{P}}_1,{\mathfrak{P}}_2,{\mathfrak{P}}_3\rangle \simeq (3,3,3)$ generated by the prime ideals lying over $\mathfrak{p}{\mathcal{O}}_{B_j}={\mathfrak{P}}_1·{\mathfrak{P}}_2·{\mathfrak{P}}_3$ for $j=6,9$. Here, we have a non-fixed-point capitulation $ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{p}\right]\rangle$ for $j=5,8$, and $ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$ for $j=6,9$. The transposition is not hidden by total capitulation and characteristic for uniform transfer kernel type $\mathrm{b}.10$. According to Theorem 9 and Corollary 4, the abelian-type invariants $\alpha =[111,111,21,21]$, and the transfer kernel type $\varkappa =\left(2100\right)$, identify six possible groups $\mathfrak{M}\simeq \langle 729,34..39\rangle$.

For ${\mathcal{N}}^{*}=7$, only three wild indices $r_j=3$, $I_j=3$, $h_3\left(B_j\right)=27$ for $j=3,4,7$ are not maximal. The TKTs are not uniform, $\varkappa =\left(2130\right)$, type $\mathrm{d}.23$, twice and $\varkappa =\left(2140\right)$, type $\mathrm{d}.25$, twice.

For ${\mathcal{N}}^{*}=10$, all tame and wild indices are maximal $I_j=27$ for $1\le j\le 10$, which implies non-uniform minimal TKTs ${\varkappa }_0=\left(2134\right)$, type $\mathrm{G}.16$, twice and ${\varkappa }_0=\left(2143\right)$, type $\mathrm{G}.19$, twice. □

Corollary 9

(Non-uniformity of the quartet for $\mathrm{III}.5$). For ${\mathcal{N}}^{*}=1$, only a sub-triplet of the quartet shares a common capitulation type $\varkappa \left(k_{\mu }\right)$, abelian-type invariants $\alpha \left(k_{\mu }\right)$, and second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$. The invariants of the fourth component differ. For ${\mathcal{N}}^{*}\in \{3,7,10\}$, only two pairs of components of the quartet share a common capitulation type $\varkappa \left(k_{\mu }\right)$, and second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, whereas the abelian-type invariants $\alpha \left(k_{\mu }\right)$ are uniform. However, the four components agree in all situations with even ${\mathcal{N}}^{*}\in \{0,2,4\}$.

Proof. 

This is an immediate consequence of Theorem 15 and Table 12. □

• In terms of capitulation targets in Corollary 2, Theorem 15 and parts of its proof are now summarized in Table 12, with transpositions in bold font.

Example 5.

The prototypes for Graph $\mathrm{III}.5$, i.e., the cases in the Theorem 15 are four  regular   situations, $c=14049$ with $\{7\leftrightarrow 223;9\}$ and ${\mathcal{N}}^{*}=0$; $c=17073$ with $\{9\leftrightarrow 271;7\}$ and ${\mathcal{N}}^{*}=2$; $c=20367$ with $\{9\leftrightarrow 73;31\}$ and ${\mathcal{N}}^{*}=1$; $c=21231$ with $\{7\leftrightarrow 337;9\}$ and ${\mathcal{N}}^{*}=3$; and the  singular   situation $c=42399$ with $\{7\leftrightarrow 673;9\}$ and ${\mathcal{N}}^{*}=4$. Here, we have distinct $(m,n)=(0,1)$, but $(\tilde{m},\tilde{n})=(1,0)$. There is also a  super-singular   prototype $c=48447$ with $\{7\leftrightarrow 769;9\}$ and ${\mathcal{N}}^{*}=4$, phenomenologically completely identical with the singular prototype, except that $(m,n)=(\tilde{m},\tilde{n})=(0,1)$. With  considerable statistic delay, there also appear super-singular prototypes ${\mathcal{N}}^{*}\in \{7,10\}$. For 37 examples see [8] (Tbl. 6.7, p. 52).

In

Table 13. Prototypes for Graph III.5.

No.	c	$\mathit{p}\leftrightarrow \mathit{q},\mathit{r}$	${\mathit{v}}^{*}$	v	$\tilde{\mathit{v}}$	$\mathit{m},\mathit{n}$	$\tilde{\mathit{m}},\tilde{\mathit{n}}$	ℓ	s	Capitulation Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	$\mathrm{14,049}$	$7\leftrightarrow 223,9$	1	2	2	$2,1$	$2,1$	1	1	$\mathrm{a}.1$	${\langle 243,28..30\rangle }^4$	$=2$
2	$\mathrm{17,073}$	$9\leftrightarrow 271,7$	2	2	2	$0,1$	$0,1$	1	1	$\mathrm{a}.3^{*}$	${\langle 81,7\rangle }^4$	$=2$
3	$\mathrm{20,367}$	$9\leftrightarrow 73,31$	1	2	2	$2,1$	$2,1$	2	2	$\mathrm{a}.2,\mathrm{a}.1$	$\langle 243,27\rangle ,{\langle 243,28..30\rangle }^3$	$=2$
4	$\mathrm{21,231}$	$7\leftrightarrow 337,9$	1	2	2	$1,1$	$1,1$	1	1	$\mathrm{a}.2,\mathrm{a}.3$	${\langle 243,27\rangle }^2,{\langle 243,25\rangle }^2$	$=2$
13	$\mathrm{42,399}$	$7\leftrightarrow 673,9$	3	3	3	$0,1$	$1,0$	1	2	$\mathrm{b}.10$	${\langle 729,37..39\rangle }^4$	$\ge 2$
16	$\mathrm{48,447}$	$7\leftrightarrow 769,9$	4	3	3	$0,1$	$0,1$	1	1	$\mathrm{b}.10$	${\langle 729,37..39\rangle }^4$	$\ge 2$
39	$\mathrm{100,503}$	$13\leftrightarrow 859,9$	3	3	3	$1,0$	$1,1$	2	1	$\mathrm{b}.10$	${\langle 729,34..36\rangle }^4$	$\ge 2$
67	$\mathrm{145,593}$	$7\leftrightarrow 2311,9$	4	3	3	$2,1$	$2,1$	1	1	$\mathrm{d}.23,\mathrm{d}.25$	${\langle 2187,250\rangle }^2,$ ${\langle 2187,251|252\rangle }^2,$	$\ge 2$
128	$\mathrm{256,669}$	$37\leftrightarrow 991,7$	6	5	3	$1,1$	$2,1$	2	1	$\mathrm{G}.16,\mathrm{G}.19$	${\left(S_4^4\right)}^2,{\left(U_5^4\right|V_6^4)}^2$	$\ge 3$

, we summarize the prototypes of graph $\mathrm{III}.5$. The data comprise ordinal number No.; conductor c of k; combined cubic residue symbol ${[p,q,r]}_3$; regularity or (super-)singularity expressed by 3-valuation $v^{*}=v_3(\#\mathrm{Cl}\left(k^{*}\right))$ of class number of absolute 3-genus field $k^{*}$; 3-valuation $v=v_3(\#\mathrm{Cl}\left(k_{pq}\right))$ and $\tilde{v}=v_3(\#\mathrm{Cl}\left({\tilde{k}}_{pq}\right))$, respectively, of the class number of critical fields $k_{pq}$ and ${\tilde{k}}_{pq}$, respectively; critical exponents $m,n$ in principal factor $A\left(k_{pq}\right)=p^m{q}^n$, $\tilde{m},\tilde{n}$ in $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, ℓ in $A\left(k_{pr}\right)=p^{\mathsf{\ell }}r$, and s in $A\left(k_{qr}\right)=q^{s}r$, respectively; capitulation type of k, second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ of k; and length ${\mathsf{\ell }}_3\left(k\right)$ of 3-class field tower of k. For abbreviation, we put the following:

• $P_7:=\langle 2187,64\rangle$, $R_4^4:=P_7-\#2;54$, $R_5^4:=P_7-\#2;57$, $R_6^4:=P_7-\#2;59$,
• $S_4^4:=R_4^4-\#1;8-\#1;3|7$, $U_5^4:=R_5^4-\#1;1-\#1;3|6$, $V_6^4:=R_6^4-\#1;6-\#1;2|6$.
• See the tables and tree diagrams in [24] (Sections 11.3 and 11.4, pp. 108–116, Tbl. 4–5, Fig. 9–11).

8.2. Category III, Graph 6

Let the combined cubic residue symbol of three primes dividing the conductor $c=pqr$ be ${[p,q,r]}_3=\{r\leftarrow p\leftrightarrow q\}$.

Proposition 10

(Quartet with 3-rank two for $\mathrm{III}.6$). For fixed $\mu \in \{1,2,3,4\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$, then the principal factor of $k_{\mu }$ is $A\left(k_{\mu }\right)=p$, and the 3-class group of $k_{\mu }$ is,

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3).$$

(56)

The unramified cyclic cubic relative extensions of $k_{\mu }$ are among the absolutely bicyclic bicubic fields $B_i$, $1\le i\le 10$. The  tame   extensions with $9\mid h_3\left(B_i\right)=(U_i:V_i)\in \{9,27\}$ are $B_i$ with $i=5,6,8,9$, since they neither contain $k_{pq}$ nor ${\tilde{k}}_{pq}$. For each μ, there are two tame extensions $B_j/k_{\mu }$, $B_{\mathsf{\ell }}/k_{\mu }$ with the following properties. The first, $B_j$ with $j\in \{6,9\}$, has the norm class group $N_{B_j/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_j\right)\right)=\langle \left[{\mathfrak{qr}}^s\right]\rangle$ with $s\in \{1,2\}$, the  cyclic   transfer kernel

$$ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$$

(57)

of order 3, and the  elementary tricyclic  3-class group ${\mathrm{Cl}}_3\left(B_j\right)=\langle \left[{\mathfrak{QR}}^s{\mathfrak{P}}_1\right],\left[{\mathfrak{QR}}^s{\mathfrak{P}}_2\right],\left[{\mathfrak{QR}}^s{\mathfrak{P}}_3\right]\rangle \simeq (3,3,3)$, generated by the classes of the prime ideals of $B_j$ over $\mathfrak{p}{\mathcal{O}}_{B_j}={\mathfrak{P}}_1{\mathfrak{P}}_2{\mathfrak{P}}_3$, $\mathfrak{q}{\mathcal{O}}_{B_j}=\mathfrak{Q}$, $\mathfrak{r}{\mathcal{O}}_{B_j}=\mathfrak{R}$. The second, $B_{\mathsf{\ell }}$ with $\mathsf{\ell }\in \{5,8\}$, has the norm class group $N_{B_{\mathsf{\ell }}/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$, transfer kernel

$$ker\left(T_{B_{\mathsf{\ell }}/k_{\mu }}\right)\ge \langle \left[{\mathfrak{qr}}^s\right]\rangle ,$$

(58)

and the 3-class group ${\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)=\langle \left[{\mathfrak{Q}}_1\right],\left[{\mathfrak{Q}}_2\right],\left[{\mathfrak{Q}}_3\right]\rangle \ge (3,3)$, generated by the classes of the prime ideals of $B_{\mathsf{\ell }}$ over $\mathfrak{q}{\mathcal{O}}_{B_{\mathsf{\ell }}}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$. The pair $(j,\mathsf{\ell })$ forms a hidden or actual  transposition   of the transfer kernel type $\varkappa \left(k_{\mu }\right)$. The remaining two $B_i>k_{\mu }$, $i\ne j$, $i\ne \mathsf{\ell }$, have the norm class group $\langle \left[\mathfrak{r}\right]\rangle$ or $\langle \left[{\mathfrak{q}}^2{\mathfrak{r}}^s\right]\rangle$, and the transfer kernel

$$ker\left(T_{B_i/k_{\mu }}\right)\ge \langle \left[\mathfrak{r}\right]\rangle ,\mathit{or}\ge \langle \left[{\mathfrak{q}}^2{\mathfrak{r}}^s\right]\rangle ,$$

providing the option of either two possible  fixed points   or a further  transposition   in the transfer kernel type $\varkappa \left(k_{\mu }\right)$. In terms of n and $\tilde{n}$ in $A\left(k_{pq}\right)=p^m{q}^n$ and $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, the ranks of the  wild   extensions are

$$r_1=r_2=r_{10}=3\mathit{iff}n\ne 0\mathit{iff}q\mid A\left(k_{pq}\right)\mathit{and}{r}_3=r_4=r_7=3\mathit{iff}\tilde{n}\ne 0\mathit{iff}q\mid A\left({\tilde{k}}_{pq}\right).$$

(59)

Proof. 

By Proposition 3, principal factors are $A\left(k_{pr}\right)=A\left({\tilde{k}}_{pr}\right)=p$, since $r\leftarrow p$. Further, by Proposition 4, $A\left(k_{\mu }\right)=p$, for all $1\le \mu \le 4$, since p is universally repelling $r\leftarrow p\to q$. Since $\mathfrak{p}=\alpha {\mathcal{O}}_{k_{\mu }}$ is a principal ideal, its class $\left[\mathfrak{p}\right]=1$ is trivial, whereas the classes $\left[\mathfrak{q}\right],\left[\mathfrak{r}\right]$ are non-trivial.

Assume the principal factors $A\left(k_{qr}\right)=q{r}^2$ and $A\left({\tilde{k}}_{qr}\right)=qr$. The parameters $m,n,\tilde{m},\tilde{n}$, proposed for all Graphs 5–9 of Category $\mathrm{III}$, decide about the rank $r_j$ of the associated principal factor matrices $M_j$ of the wild bicyclic bicubic fields $B_j$, $j\in \{1,2,3,4,7,10\}$. As usual, we perform row operations on these matrices:

$M_1=\left(\begin{array}{ccc}1& 0& 0\\ m& n& 0\\ 1& 0& 0\\ 0& 1& 2\end{array}\right)$, $M_2=\left(\begin{array}{ccc}1& 0& 0\\ m& n& 0\\ 1& 0& 0\\ 0& 1& 1\end{array}\right)$, $M_{10}=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\\ m& n& 0\end{array}\right)$.

For $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $M_1$ leads to the decisive pivot element $-2n$ in the last column; similarly, for $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $M_2$ leads to $-n$; and similarly, for $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $M_{10}$ leads to n in the middle column. So, $r_1=r_2=r_{10}=3$ iff $n\ne 0$, by viewing the pivot elements modulo 3. Next:

$M_3=\left(\begin{array}{ccc}1& 0& 0\\ \tilde{m}& \tilde{n}& 0\\ 1& 0& 0\\ 0& 1& 2\end{array}\right)$, $M_4=\left(\begin{array}{ccc}1& 0& 0\\ \tilde{m}& \tilde{n}& 0\\ 1& 0& 0\\ 0& 1& 1\end{array}\right)$, $M_7=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\\ \tilde{m}& \tilde{n}& 0\end{array}\right)$.

For $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $M_3$ leads to $-2\tilde{n}$; similarly, for $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_4$ leads to $-\tilde{n}$; and similarly, for $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ leads to $\tilde{n}$ in the middle column. So, $r_3=r_4=r_7=3$ iff $\tilde{n}\ne 0$.

In the regular case $h_3\left(k_{pq}\right)=h_3\left({\tilde{k}}_{pq}\right)=9$, where $(m,n)=(\tilde{m},\tilde{n})$, the condition $n\ne 0$, that is, $q\mid A\left(k_{pq}\right)$, is certainly satisfied when $\mathcal{P}=2$ or equivalently $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{pq}\right)/k_{pq})\simeq \langle 9,2\rangle$, according to Theorem 4. However, when $\mathcal{P}=1$ or equivalently $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{pq}\right)/k_{pq})\simeq \langle 27,4\rangle$, then we may either have $q\mid A\left(k_{pq}\right)$ and still $n\ne 0$, or $p\mid A\left(k_{pq}\right)$, $n=0$, with completely different consequence $r_1=r_2=r_{10}=r_3=r_4=r_7=2$. In the singular and super-singular cases, both pairs of parameters, $(m,n)$ and $(\tilde{m},\tilde{n})$, or more precisely, only n and $\tilde{n}$, must be taken into consideration separately. See also the proof of Theorem 16. □

In terms of capitulation targets in Corollary 2, Theorem 16 and parts of its proof are now summarized in

Table 14. Norm class groups and minimal transfer kernels for Graph III.6.

Base	${\mathit{k}}_1$				${\mathit{k}}_2$				${\mathit{k}}_3$				${\mathit{k}}_4$
Ext	${\mathit{B}}_1$	${\mathit{B}}_5$	${\mathit{B}}_6$	${\mathit{B}}_7$	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$	${\mathit{B}}_4$	${\mathit{B}}_6$	${\mathit{B}}_8$	${\mathit{B}}_{10}$
NCG	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{qr}$	$\mathfrak{r}$	$\mathfrak{qr}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{r}$
TK	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{r}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$	$\mathfrak{r}$
$\varkappa$	$\mathbf{4}$	$\mathbf{3}$	$\mathbf{2}$	$\mathbf{1}$	$\mathbf{2}$	$\mathbf{1}$	$\mathbf{4}$	$\mathbf{3}$	1	$\mathbf{3}$	$\mathbf{2}$	4	1	$\mathbf{3}$	$\mathbf{2}$	4

, with transpositions in bold font.

Theorem 16

(Second 3-class group for $\mathrm{III}.6$). To identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $1\le \mu \le 4$, let the principal factor of $k_{pq}$ and ${\tilde{k}}_{pq}$, respectively, be $A\left(k_{pq}\right)=p^m{q}^n$ and $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, respectively, and additionally, assume the regular situation where both ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq {\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)\simeq (3,3)$ are elementary bicyclic, whence $(m,n)=(\tilde{m},\tilde{n})$.

Then there are several  minimal transfer kernel types   (mTKT) ${\varkappa }_0$ of $k_{\mu }$, $1\le \mu \le 4$, and other possible capitulation types in ascending order ${\varkappa }_0<\varkappa <{\varkappa }^{\prime }<{\varkappa }^{″}$, ending in the mandatory ${\varkappa }^{″}=\left(2000\right)$, type $\mathrm{a}.3^{*}$, either ${\varkappa }_0=\left(2134\right)$, type $\mathrm{G}.16$, $\varkappa =\left(2130\right)$, type $\mathrm{d}.23$, ${\varkappa }^{\prime }=\left(2100\right)$, type $\mathrm{b}.10$, or ${\varkappa }_0=\left(2143\right)$, type $\mathrm{G}.19$, $\varkappa =\left(2140\right)$, type $\mathrm{d}.25$, and again ${\varkappa }^{\prime }=\left(2100\right)$, type $\mathrm{b}.10$.

In the  regular   situation, the second 3-class group is $\mathfrak{M}\simeq$

$$\left\{\begin{array}{cc}{\langle 81,7\rangle }^4,\alpha ={[111,11,11,11]}^4,\varkappa ={\left(2000\right)}^4\hfill & \mathit{if}n\ne 0,\mathcal{N}=1,\hfill \\ {\langle 729,34..39\rangle }^4,\alpha ={[111,111,21,21]}^4,\varkappa ={\left(2100\right)}^4\hfill & \mathit{if}n=0,\mathcal{N}=2,\hfill \\ {\langle 729,43\rangle }^2,{\langle 729,42\rangle }^2,\alpha ={[111,111,22,21]}^4,\varkappa ={\left(2140\right)}^2,{\left(2130\right)}^2\hfill & \mathit{if}n=0,\mathcal{N}=3,\hfill \\ {\langle 2187,71\rangle }^2,{\langle 2187,69\rangle }^2\alpha ={[111,111,22,22]}^4,\varkappa ={\left(2143\right)}^2,{\left(2134\right)}^2\hfill & \mathit{if}n=0,\mathcal{N}=4,\hfill \end{array}\right.$$

(60)

where $\mathcal{N}:=\#\{1\le w\le 10\mid k_{\mu }<B_w,I_w=27\}$. Only in the first case, the 3-class field tower has certainly the group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$ and length ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$, otherwise ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$ cannot be excluded, even if $d_2\left(\mathfrak{M}\right)\le 4$.

In the  singular   situation, the second 3-class group is $\mathfrak{M}\simeq$

$$\left\{\begin{array}{cc}{\langle 2187,251|252\rangle }^2,{\langle 2187,250\rangle }^2,\alpha ={[111,111,32,21]}^4,\varkappa ={\left(2140\right)}^2,{\left(2130\right)}^2\hfill & \mathit{if}n=\tilde{n}=0,\mathcal{N}=3.\hfill \end{array}\right.$$

(61)

In the  super-singular   situation, no statement is possible, since the order of $\mathfrak{M}$ may increase unboundedly.

Proof. 

In the regular situation ${\mathrm{Cl}}_3\left(k_{pq}\right)={\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)=(3,3)$, exponents $(m,n)$ and $(\tilde{m},\tilde{n})$ of principal factors $A\left(k_{pq}\right)=p^m{q}^n$ and $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$ are equal. Let $\mathcal{P}$ be the number of primes dividing $A\left(k_{pq}\right)$. According to the proof of Proposition 10, ranks $r_w$ and indices $I_w:=(U_w:V_w)$ of subfield units for wild extensions are given by $r_w=3$, $I_w=3$ iff $n\ne 0$, for $w\in \{1,2,10\}$, and $r_w=3$, $I_w=3$ iff $\tilde{n}\ne 0$, for $w\in \{3,4,7\}$, in particular, certainly for $\mathcal{P}=2$.

This implies 3-class numbers $h_3\left(B_w\right)=h_3\left(k_{pq}\right)=h_3\left({\tilde{k}}_{pq}\right)=9$ and 3-class groups ${\mathrm{Cl}}_3\left(B_w\right)\simeq (3,3)$, for $w\in \{1,2,3,4,7,10\}$, whenever $n\ne 0$, i.e., $q\mid A\left(k_{pq}\right)$, a remarkable distinction of the prime q against the primes $p,r$. We point out that this can occur not only for $\mathcal{P}=2$, but also for $\mathcal{P}=1$, provided that $A\left(k_{pq}\right)=q$, $n=1$, and not $A\left(k_{pq}\right)=p$, $m=1$.

Indices of tame extensions with $9\mid h_3\left(B_w\right)=I_w\in \{9,27\}$ and $r_w=2$ are non-uniform: corresponding to a unique elementary tricyclic ${\mathrm{Cl}}_3\left(B_w\right)\simeq (3,3,3)$, we must have $I_w=27$ for $w\in \{6,9\}$ with the norm class group $N_{B_w/k_{\mu }}{\mathrm{Cl}}_3\left(B_w\right)$ either $\langle \left[\mathfrak{qr}\right]\rangle$ or $\langle \left[{\mathfrak{qr}}^2\right]\rangle$, but corresponding to the remaining bicyclic ${\mathrm{Cl}}_3\left(B_w\right)\simeq (3,3)$, the index $I_w=9$ takes the minimal value for $w\in \{5,8\}$ with the norm class group $N_{B_w/k_{\mu }}{\mathrm{Cl}}_3\left(B_w\right)=\langle \left[\mathfrak{q}\right]\rangle$. Thus $\mathcal{N}=1$ and the resulting Artin pattern $\alpha =[111,11,11,11]$ uniquely identifies the group $\mathfrak{G}=\mathfrak{M}\simeq \langle 81,7\rangle$ of maximal class.

Now, we come to $n=0$, whence necessarily $\mathcal{P}=1$. Then, $r_w=2$ and $I_w\in \{9,27\}$ for the wild extensions $w\in \{1,2,3,4,7,10\}$. Indices of tame extensions now become uniform, corresponding to a pair of elementary tricyclic ${\mathrm{Cl}}_3\left(B_w\right)\simeq (3,3,3)$, which enforces $I_w=27$ for $w\in \{5,6,8,9\}$, i.e., $\mathcal{N}\ge 2$. The number $\mathcal{N}$ of maximal unit indices decides about the group $\mathfrak{M}$: If $\mathcal{N}=2$, then for all $w\in \{1,2,3,4,7,10\}$: $I_w=9$, $h_3\left(B_w\right)=3·h_3\left(k_{pq}\right)=27$, and ${\mathrm{Cl}}_3\simeq (9,3)$, according to the laws for 3-groups of coclass $\ge 2$ [25] (pp. 289–292). The Artin pattern $\alpha =[111,111,21,21]$ identifies the possible groups $\mathfrak{M}\simeq \langle 729,34..39\rangle$. If $\mathcal{N}=3$, then $I_w=27$ for $w\in \{3,4,7\}$, but $I_w=9$ for $w\in \{1,2,10\}$. The Artin pattern $\alpha =[111,111,22,21]$ together with $\varkappa ={\left(2140\right)}^2$, $\varkappa ={\left(2130\right)}^2$, according to Table 14, identifies the possible groups $\mathfrak{M}\simeq {\langle 2187,251|252\rangle }^2,{\langle 2187,250\rangle }^2$ of coclass 2. If $\mathcal{N}=4$, then for all $w\in \{1,2,3,4,7,10\}$: $I_w=27$, $h_3\left(B_w\right)=9·h_3\left(k_{pq}\right)=81$, and ${\mathrm{Cl}}_3\simeq (9,9)$. The Artin pattern $\alpha =[111,111,22,22]$ together with $\varkappa ={\left(2143\right)}^2$, $\varkappa ={\left(2134\right)}^2$, according to Table 14, identifies the possible groups $\mathfrak{M}\simeq {\langle 2187,71\rangle }^2,{\langle 2187,69\rangle }^2$ of coclass 3. □

Corollary 10

(Non-uniformity of the quartet for $\mathrm{III}.6$). Only for $\mathcal{N}\le 2$, the components of the quartet, all with 3-rank two, share a common capitulation type $\varkappa \left(k_{\mu }\right)$, common abelian-type invariants $\alpha \left(k_{\mu }\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, for $1\le \mu \le 4$. For $\mathcal{N}\ge 3$, the quartet splits into two sub-doublets, and thus becomes non-uniform.

Proof. 

This is an immediate consequence of Theorem 16 and Table 14. □

Example 6.

For 31 examples see [8] (Tbl. 6.4, p. 49). Prototypes for Graph $\mathrm{III}.6$, that is, minimal conductors for each scenario in Theorem 16, are the following.

There are the  regular   cases $c=8541$ with symbol $\{9\leftrightarrow 73\to 13\}$, $(m,n)=(1,2)$; $c=9373$ with symbol $\{103\leftrightarrow 13\to 7\}$, $(m,n)=(1,1)$; $c=\mathrm{56,329}$ with symbol $\{619\leftrightarrow 13\to 7\}$, $(m,n)=(0,1)$, all uniformly with $\mathfrak{G}=\mathfrak{M}={\langle 81,7\rangle }^4$, in contrast to $c=\mathrm{142,219}$ with symbol $\{19\leftrightarrow 577\to 13\}$, $(m,n)=(1,0)$, and uniform $\mathfrak{M}={\langle 729,37..39\rangle }^4$; $c=\mathrm{152,893}$ with symbol $\{13\leftrightarrow 619\to 19\}$, $(m,n)=(1,0)$, and uniform $\mathfrak{M}={\langle 729,34..36\rangle }^4$; $c=\mathrm{163,681}$ with symbol $\{67\leftrightarrow 349\to 7\}$, $(m,n)=(1,0)$, and non-uniform $\mathfrak{M}={\langle 729,42\rangle }^2,{\langle 729,43\rangle }^2$; $c=\mathrm{193,059}$ with symbol $\{1129\leftrightarrow 19\to 9\}$, $(m,n)=(1,0)$, and non-uniform $\mathfrak{M}={\langle 2187,69\rangle }^2,{\langle 2187,71\rangle }^2$ with two distinct minimal transfer kernel types.

Further, the  singular   cases $c=78,169$ with symbol $\{859\leftrightarrow 13\to 7\}$, $(m,n)=(1,0)$, $(\tilde{m},\tilde{n})=(1,1)$, and non-uniform $\mathfrak{M}={\langle 2187,250\rangle }^2,{\langle 2187,251|252\rangle }^2$; $c=142,947$ with symbol $\{9\leftrightarrow 2269\to 7\}$, $(m,n)=(1,0)$, $(\tilde{m},\tilde{n})=(0,1)$, and uniform $\mathfrak{M}={\langle 2187,253\rangle }^4$.

Finally, the  super-singular   cases $c=\mathrm{102,277}$ with symbol $\{769\leftrightarrow 7\to 19\}$, $(m,n)=(\tilde{m},\tilde{n})=(0,1)$, and uniform $\mathfrak{M}={\langle 729,37..39\rangle }^4$; $c=\mathrm{199,171}$ with symbol $\{7\leftrightarrow 769\to 37\}$, $(m,n)=(\tilde{m},\tilde{n})=(1,0)$, and uniform $\mathfrak{M}={\langle 6561,693..698\rangle }^4$.

In

Table 15. Prototypes for Graph III.6.

No.	c	$\mathit{r}\leftarrow \mathit{p}\leftrightarrow \mathit{q}$	${\mathit{v}}^{*}$	v	$\mathit{m},\mathit{n}$	$\tilde{\mathit{v}}$	$\tilde{\mathit{m}},\tilde{\mathit{n}}$	Capitulation   Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	8541	$13\leftarrow 73\leftrightarrow 9$	1	2	$1,2$	2	$1,2$	$\mathrm{a}.3^{*}$	${\langle 81,7\rangle }^4$	$=2$
2	9373	$7\leftarrow 13\leftrightarrow 103$	1	2	$1,1$	2	$1,1$	$\mathrm{a}.3^{*}$	${\langle 81,7\rangle }^4$	$=2$
20	$\mathrm{56,329}$	$7\leftarrow 13\leftrightarrow 619$	2	2	$0,1$	2	$0,1$	$\mathrm{a}.3^{*}$	${\langle 81,7\rangle }^4$	$=2$
29	$\mathrm{78,169}$	$7\leftarrow 13\leftrightarrow 859$	3	3	$1,0$	3	$1,1$	$\mathrm{d}.23$	${\langle 2187,250\rangle }^2$	$\ge 2$
								$\mathrm{d}.25$	${\langle 2187,251|252\rangle }^2$	$\ge 2$
34	$\mathrm{102,277}$	$19\leftarrow 7\leftrightarrow 769$	4	3	$0,1$	3	$0,1$	$\mathrm{b}.10$	${\langle 729,37..39\rangle }^4$	$\ge 2$
52	$\mathrm{142,519}$	$13\leftarrow 577\leftrightarrow 19$	2	2	$1,0$	2	$1,0$	$\mathrm{b}.10$	${\langle 729,37..39\rangle }^4$	$\ge 2$
54	$\mathrm{142,947}$	$7\leftarrow 2269\leftrightarrow 9$	3	3	$1,0$	3	$0,1$	$\mathrm{b}.10$	${\langle 2187,253\rangle }^4$	$\ge 2$
56	$\mathrm{152,893}$	$19\leftarrow 619\leftrightarrow 13$	2	2	$1,0$	2	$1,0$	$\mathrm{b}.10$	${\langle 729,34..36\rangle }^4$	$\ge 2$
58	$\mathrm{163,681}$	$7\leftarrow 349\leftrightarrow 67$	2	2	$1,0$	2	$1,0$	$\mathrm{d}.23$	${\langle 729,42\rangle }^2$	$\ge 2$
								$\mathrm{d}.25$	${\langle 729,43\rangle }^2$	$\ge 2$
71	$\mathrm{193,059}$	$9\leftarrow 19\leftrightarrow 1129$	2	2	$1,0$	2	$1,0$	$\mathrm{G}.16$	${\langle 2187,69\rangle }^2$	$\ge 2$
								$\mathrm{G}.19$	${\langle 2187,71\rangle }^2$	$\ge 2$
75	$\mathrm{199,171}$	$37\leftarrow 769\leftrightarrow 7$	4	3	$1,0$	3	$1,0$	$\mathrm{b}.10$	$\langle 6561,693..$698〉4	$\ge 3$

, we summarize the prototypes of Graph $\mathrm{III}.6$ in the same way as in Table 13.

8.3. Category III, Graph 7

Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 7 of Category $\mathrm{III}$ with combined cubic residue symbol ${[p,q,r]}_3=\{r\to p\leftrightarrow q\}$.

Proposition 11

(Quartet with 3-rank two for $\mathrm{III}.7$). For fixed $\mu \in \{1,2,3,4\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$. Under the normalizing assumption $A\left(k_{qr}\right)=q{r}^2$, $A\left({\tilde{k}}_{qr}\right)=qr$, the principal factors of $k_{\mu }$ are

$$A\left(k_1\right)=A\left(k_3\right)=qr\mathit{and}A\left(k_2\right)=A\left(k_4\right)=q{r}^2,$$

(62)

and the 3-class group of $k_{\mu }$ is

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{p}\right],\left[\mathfrak{q}\right]\rangle =\langle \left[\mathfrak{p}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3).$$

(63)

The unramified cyclic cubic relative extensions of $k_{\mu }$ are among the absolutely bicyclic bicubic fields $B_i$, $1\le i\le 10$. The  tame   extensions with $9\mid h_3\left(B_i\right)=(U_i:V_i)\in \{9,27\}$ are $B_i$ with $i=5,6,8,9$, since they neither contain $k_{pq}$ nor ${\tilde{k}}_{pq}$. For each μ, there are two tame extensions $B_j/k_{\mu }$, $B_{\mathsf{\ell }}/k_{\mu }$ with the following properties. The first, $B_j$ with $j\in \{6,9\}$, has the norm class group $N_{B_j/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_j\right)\right)=\langle \left[\mathfrak{p}\right]\rangle$, the transfer kernel

$$ker\left(T_{B_j/k_{\mu }}\right)\ge \langle \left[\mathfrak{q}\right]\rangle ,$$

(64)

and the 3-class group ${\mathrm{Cl}}_3\left(B_j\right)=\langle \left[{\mathfrak{P}}_1\right],\left[{\mathfrak{P}}_2\right],\left[{\mathfrak{P}}_3\right]\rangle \ge (3,3)$, generated by the classes of the prime ideals of $B_j$ over $\mathfrak{p}{\mathcal{O}}_{B_j}={\mathfrak{P}}_1{\mathfrak{P}}_2{\mathfrak{P}}_3$. The second, $B_{\mathsf{\ell }}$ with $\mathsf{\ell }\in \{5,8\}$, has the norm class group $N_{B_{\mathsf{\ell }}/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$, the  cyclic   transfer kernel

$$ker\left(T_{B_{\mathsf{\ell }}/k_{\mu }}\right)=\langle \left[\mathfrak{p}\right]\rangle$$

(65)

of order 3, and the  elementary tricyclic  3-class group ${\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)=\langle \left[{\mathfrak{Q}}_1\right],\left[{\mathfrak{Q}}_2\right],\left[{\mathfrak{Q}}_3\right]\rangle \simeq (3,3,3)$, generated by the classes of the prime ideals of $B_{\mathsf{\ell }}$ over $\mathfrak{q}{\mathcal{O}}_{B_{\mathsf{\ell }}}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$. The pair $(j,\mathsf{\ell })$ forms a hidden or actual  transposition   of the transfer kernel type $\varkappa \left(k_{\mu }\right)$. The remaining two $B_i>k_{\mu }$, $i\ne j$, $i\ne \mathsf{\ell }$, have the norm class group $\langle \left[\mathfrak{pq}\right]\rangle$ or $\langle \left[{\mathfrak{pq}}^2\right]\rangle$, and the transfer kernel

$$ker\left(T_{B_i/k_{\mu }}\right)\ge \langle \left[\mathfrak{q}\right]\rangle ,$$

providing the option of two possible  repetitions   in the transfer kernel type $\varkappa \left(k_{\mu }\right)$.

• In terms of n and $\tilde{n}$ in $A\left(k_{pq}\right)=p^m{q}^n$ and $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, the ranks of the  wild   extensions are

  $$r_1=r_2=r_{10}=3\mathit{iff}m\ne 0\mathit{iff}p\mid A\left(k_{pq}\right)\mathit{and}{r}_3=r_4=r_7=3\mathit{iff}\tilde{m}\ne 0\mathit{iff}p\mid A\left({\tilde{k}}_{pq}\right).$$

  (66)

Proof. 

By Proposition 3, the symbol $r\to p$ implies principal factors $A\left(k_{pr}\right)=A\left({\tilde{k}}_{pr}\right)=r$.

We assume principal factors $A\left(k_{\mu }\right)=p^{x_{\mu }}{q}^{y_{\mu }}{r}^{z_{\mu }}$, for $1\le \mu \le 4$, and $A\left(k_{qr}\right)=q{r}^2$, $A\left({\tilde{k}}_{qr}\right)=qr$.

We generally have the tame matrix ranks $r_5=r_6=r_8=r_9=2$, and draw conclusions by explicit calculations. For these bicyclic bicubic fields $B_j$, $j\in \{5,6,8,9\}$, the rank $r_j$ is calculated with row operations on the associated principal factor matrices $M_j$:

$M_5=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_3& y_3& z_3\\ 1& 0& 0\\ 0& 1& 1\end{array}\right)$, $M_6=\left(\begin{array}{ccc}{x}_1& y_1& z_1\\ x_4& y_4& z_4\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$, $M_8=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ x_4& y_4& z_4\\ 1& 0& 0\\ 0& 1& 2\end{array}\right)$, $M_9=\left(\begin{array}{ccc}{x}_2& y_2& z_2\\ x_3& y_3& z_3\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$.

For $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $M_5$ leads to the decisive pivot elements $z_1-y_1$ and $z_3-y_3$; similarly, for $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $M_6$ leads to $x_1$ and $x_4$; similarly, for $B_8=k_2{k}_4{k}_p{k}_{qr}$, $M_8$ leads to $z_2-2y_2$ and $z_4-2y_4$; and similarly, for $B_9=k_2{k}_3{k}_q{k}_{pr}$, $M_9$ leads to $x_2$ and $x_3$. So, $r_5=r_6=2$ implies $z_1=y_1$, $z_3=y_3$, $x_1=x_4=0$, and $r_8=r_9=2$ implies $z_2=2y_2$, $z_4=2y_4$, $x_2=x_3=0$, i.e., $A\left(k_1\right)=A\left(k_3\right)=qr$ and $A\left(k_2\right)=A\left(k_4\right)=q{r}^2$.

A consequence of these principal factors is the coincidence of the subgroups of ${\mathrm{Cl}}_3\left(k_{\mu }\right)$ generated by the classes $\left[\mathfrak{q}\right]$ and $\left[\mathfrak{r}\right]$ in $k_{\mu }$, $\mu =1,\dots ,4$. By Corollary 3,

since $\mathfrak{p}$ is principal ideal in $k_p$, the class $\left[\mathfrak{p}\right]$ capitulates in $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$ and $B_8=k_2{k}_4{k}_p{k}_{qr}$;

since $\mathfrak{q}$ is principal ideal in $k_q$, the class $\left[\mathfrak{q}\right]$ capitulates in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$;

since $\mathfrak{r}$ is principal ideal in $k_r$, the class $\left[\mathfrak{r}\right]$, and thus $\left[\mathfrak{q}\right]$, capitulates in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$ and $B_{10}=k_3{k}_4{k}_r{k}_{pq}$.

Moreover, since $\mathfrak{r}$ is principal ideal in $k_{pr}$ and ${\tilde{k}}_{pr}$, the class $\left[\mathfrak{r}\right]$, and thus $\left[\mathfrak{q}\right]$, also capitulates in $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, and $B_9=k_2{k}_3{k}_q{k}_{pr}$, by Proposition 2.

The parameters $m,n,\tilde{m},\tilde{n}$, proposed for all Graphs 5–9 of Category $\mathrm{III}$, decide about the rank $r_j$ of the associated principal factor matrices $M_j$ of the wild bicyclic bicubic fields $B_j$, $j\in \{1,2,3,4,7,10\}$. As usual, we perform row operations on these matrices:

$M_1=\left(\begin{array}{ccc}0& 1& 1\\ m& n& 0\\ 0& 0& 1\\ 0& 1& 2\end{array}\right)$, $M_2=\left(\begin{array}{ccc}0& 1& 2\\ m& n& 0\\ 0& 0& 1\\ 0& 1& 1\end{array}\right)$, $M_{10}=\left(\begin{array}{ccc}0& 1& 1\\ 0& 1& 2\\ 0& 0& 1\\ m& n& 0\end{array}\right)$.

For $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $M_1$ leads to the decisive pivot element m in the first column; similarly, for $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $M_2$ leads to m; and similarly, for $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $M_{10}$ leads to m in the first column. So, $r_1=r_2=r_{10}=3$ iff $m\ne 0$. Next, we consider:

$M_3=\left(\begin{array}{ccc}0& 1& 1\\ \tilde{m}& \tilde{n}& 0\\ 0& 0& 1\\ 0& 1& 2\end{array}\right)$, $M_4=\left(\begin{array}{ccc}0& 1& 2\\ \tilde{m}& \tilde{n}& 0\\ 0& 0& 1\\ 0& 1& 1\end{array}\right)$, $M_7=\left(\begin{array}{ccc}0& 1& 1\\ 0& 1& 2\\ 0& 0& 1\\ \tilde{m}& \tilde{n}& 0\end{array}\right)$.

For $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $M_3$ leads to $\tilde{m}$; similarly, for $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_4$ leads to $\tilde{m}$; and similarly, for $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ leads to $\tilde{m}$ in the first column. So, $r_3=r_4=r_7=3$ iff $\tilde{m}\ne 0$.

Since r splits in $k_p$, it also splits in $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$.

Since q splits in $k_p$, it also splits in $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$.

Since p splits in $k_q$, it also splits in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$. □

• In terms of capitulation targets in Corollary 2, Theorem 17 and parts of its proof are now summarized in

  Table 16. Norm class groups and minimal transfer kernels for Graph III.7.

  Base	${\mathit{k}}_1$				${\mathit{k}}_2$				${\mathit{k}}_3$				${\mathit{k}}_4$
  Ext	${\mathit{B}}_1$	${\mathit{B}}_5$	${\mathit{B}}_6$	${\mathit{B}}_7$	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$	${\mathit{B}}_4$	${\mathit{B}}_6$	${\mathit{B}}_8$	${\mathit{B}}_{10}$
  NCG	${\mathfrak{pq}}^2$	$\mathfrak{q}$	$\mathfrak{p}$	$\mathfrak{pq}$	${\mathfrak{pq}}^2$	$\mathfrak{pq}$	$\mathfrak{q}$	$\mathfrak{p}$	${\mathfrak{pq}}^2$	$\mathfrak{q}$	$\mathfrak{p}$	$\mathfrak{pq}$	${\mathfrak{pq}}^2$	$\mathfrak{p}$	$\mathfrak{q}$	$\mathfrak{pq}$
  TK	$\mathfrak{q}$	$\mathfrak{p}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{p}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{p}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{q}$	$\mathfrak{p}$	$\mathfrak{q}$
  $\varkappa$	2	$\mathbf{3}$	$\mathbf{2}$	2	3	3	$\mathbf{4}$	$\mathbf{3}$	2	$\mathbf{3}$	$\mathbf{2}$	2	3	$\mathbf{3}$	$\mathbf{2}$	3

  with transpositions in bold font.

Theorem 17

(Second 3-class group for $\mathrm{III}.7$). Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 7 of Category $\mathrm{III}$ with combined cubic residue symbol ${[p,q,r]}_3=\{q\leftrightarrow p\to r\}$.

Then, the  minimal transfer kernel type   (mTKT) of $k_{\mu }$, $1\le \mu \le 4$, is ${\varkappa }_0=\left(2111\right)$, type $\mathrm{H}.4$, and the other possible capitulation types in ascending order ${\varkappa }_0<{\varkappa }^{\prime }<{\varkappa }^{″}<{\varkappa }^{‴}$ are ${\varkappa }^{\prime }=\left(2110\right)$, type $\mathrm{d}.19$; ${\varkappa }^{″}=\left(2100\right)$, type $\mathrm{b}.10$; and ${\varkappa }^{‴}=\left(2000\right)$, type $\mathrm{a}.3^{*}$.

To identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $1\le \mu \le 4$, let the  decisive principal factors   be $A\left(k_{pq}\right)=p^m{q}^n$, $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, and additionally, assume the  regular situation  where both ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq {\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)\simeq (3,3)$ are elementary bicyclic, whence $(m,n)=(\tilde{m},\tilde{n})$. Then,

$$\mathfrak{M}\simeq \left\{\begin{array}{cc}\langle 81,7\rangle ,\alpha =[111,11,11,11],\varkappa =\left(2000\right)\hfill & \mathit{if}m\ne 0,\mathcal{N}=1,\hfill \\ \langle 729,34..39\rangle ,\alpha =[111,111,21,21],\varkappa =\left(2100\right)\hfill & \mathit{if}m=0,\mathcal{N}=2,\hfill \\ \langle 729,41\rangle ,\alpha =[111,111,22,21],\varkappa =\left(2110\right)\hfill & \mathit{if}m=0,\mathcal{N}=3,\hfill \\ \langle 2187,65|67\rangle ,\alpha =[111,111,22,22],\varkappa =(2111)\hfill & \mathit{if}m=0,\mathcal{N}=4,\hfill \end{array}\right.$$

(67)

where $\mathcal{N}:=\#\{1\le j\le 10\mid k_{\mu }<B_j,I_j=27\}$. Only in the leading row has the 3-class field tower warranted the group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$, with length ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$. Otherwise, the tower length ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$ cannot be excluded, even if $d_2\left(\mathfrak{M}\right)\le 4$.

In  (super-)singular situations, the group $\mathfrak{M}$ must be of coclass $\mathrm{cc}\left(\mathfrak{M}\right)\ge 2$, and capitulation of type ${\varkappa }^{‴}=\left(2000\right)$ is impossible.

Proof. 

We know that the tame ranks are $r_5=r_6=r_8=r_9=2$, and thus $I_5,I_6,I_8,I_9\in \{9,27\}$, in particular, $I_5=I_8=27$, whence certainly $\mathcal{N}\ge 1$. Further, the wild ranks are $r_1=r_2=r_{10}=3$ iff $m\ne 0$, and $r_3=r_4=r_7=3$ iff $\tilde{m}\ne 0$.

In the regular situation where the 3-class groups of $k_{pq}$ and ${\tilde{k}}_{pq}$ are elementary bicyclic, tight bounds arise for the abelian-quotient invariants $\alpha$ of the group $\mathfrak{M}$:

The first scenario, $m\ne 0$, is equivalent to $\mathcal{N}=1$, with wild ranks $h_3\left(B_j\right)=h_3\left(k_{pq}\right)=9$, for $j=1,2,10$, $h_3\left(B_j\right)=h_3\left({\tilde{k}}_{pq}\right)=9$, for $j=3,4,7$, and tame ranks $h_3\left(B_j\right)=I_j=9$, for $j=6,9$, $h_3\left(B_j\right)=I_j=27$, for $j=5,8$, that is $\alpha =[111,11,11,11]$ and consequently $\varkappa =\left(2000\right)$, since $\langle 81,7\rangle$ is unique with this $\alpha$.

The other three scenarios share $m=0$, and an explicit transposition between $B_5$, $B_6$; $B_5$, $B_9$, and $B_6$, $B_8$; and $B_8$, $B_9$, respectively, giving rise to $\varkappa =(21\ast \ast )$, and common $h_3\left(B_j\right)=I_j=27$, for $j=5,6,8,9$, implying $\alpha =[111,111,\ast ,\ast ]$.

The second scenario with $\mathcal{N}=2$ is supplemented by $I_j=9$, $h_3\left(B_j\right)=3·h_3\left(k_{pq}\right)=27$, for $j=1,2,10$. $I_j=9$, $h_3\left(B_j\right)=3·h_3\left({\tilde{k}}_{pq}\right)=27$, for $j=3,4,7$, giving rise to $\alpha =[111,111,21,21]$, $\varkappa =\left(2100\right)$, characteristic for $\langle 729,34..39\rangle$ (Cor. 4).

The third scenario with $\mathcal{N}=3$ is supplemented by $I_j=27$, $h_3\left(B_j\right)=9·h_3\left(k_{pq}\right)=81$, for $j=1,2,10$, but still $I_j=9$, $h_3\left(B_j\right)=3·h_3\left({\tilde{k}}_{pq}\right)=27$, for $j=3,4,7$, giving rise to $\alpha =[111,111,22,21]$, $\varkappa =\left(2110\right)$, characteristic for $\langle 729,41\rangle$.

The fourth scenario with $\mathcal{N}=4$ is supplemented by $I_j=27$, $h_3\left(B_j\right)=9·h_3\left(k_{pq}\right)=81$, for $j=1,2,10$, $I_j=27$, $h_3\left(B_j\right)=9·h_3\left(k_{pq}\right)=81$, for $j=3,4,7$, giving rise to $\alpha =[111,111,22,22]$, $\varkappa =\left(2111\right)$, characteristic for either $\langle 2187,65|67\rangle$ or $\langle 6561,714..719|738..743\rangle$ with coclass $\mathrm{cc}=3$. If $d_2\left(\mathfrak{M}\right)=5$, then the tower length must be ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$. For this minimal capitulation type H.4, $\varkappa =\left(2111\right)$, all transfer kernels are cyclic of order 3, and the minimal unit norm indices correspond to maximal subfield unit indices.

In (super-)singular situations, the 3-class groups of $k_{pq}$ and ${\tilde{k}}_{pq}$ are non-elementary bicyclic, and even in the simplest case, $m\ne 0$, $\tilde{m}\ne 0$, we have $I_j=3$, $27\mid h_3\left(B_j\right)=h_3\left(k_{pq}\right)$, for $j=1,2,10$, $I_j=3$, $27\mid h_3\left(B_j\right)=h_3\left({\tilde{k}}_{pq}\right)$, for $j=3,4,7$, which prohibits the occurrence of abelian-type invariants $\left(11\right)$, required for 3-groups of coclass $\mathrm{cc}\left(\mathfrak{M}\right)=1$ (maximal class). □

Corollary 11

(Uniformity of the quartet for $\mathrm{III}.7$). The components of the quartet, all with 3-rank two, share a common capitulation type $\varkappa \left(k_{\mu }\right)$, common abelian-type invariants $\alpha \left(k_{\mu }\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, for $1\le \mu \le 4$.

Proof. 

This is a consequence of Theorem 17 and Table 16. □

Example 7.

For 34 examples see [8] (Tbl. 6.5, p. 50). We have found prototypes for Graph $\mathrm{III}.7$ in the form of minimal conductors for each scenario in Theorem 17 as follows. There are  regular   cases: $c=4599$ with symbol $\{9\leftrightarrow 73\leftarrow 7\}$, $v^{*}=1$, and $\mathfrak{G}=\mathfrak{M}={\langle 81,7\rangle }^4$; $c=\mathrm{31,707}$ with symbol $\{9\leftrightarrow 271\leftarrow 13\}$, $v^{*}=2$, and $\mathfrak{G}=\mathfrak{M}={\langle 81,7\rangle }^4$; $c=\mathrm{76,741}$ with symbol $\{577\leftrightarrow 19\leftarrow 7\}$, $v^{*}=2$, and $\mathfrak{M}={\langle 2187,65|67\rangle }^4$ of elevated coclass 3; and $c=\mathrm{90,243}$ with symbol $\{271\leftrightarrow 9\leftarrow 37\}$, $v^{*}=2$, and $\mathfrak{M}={\langle 729,41\rangle }^4$. There is also a  singular   case $c=\mathrm{61,243}$ with symbol $\{673\leftrightarrow 7\leftarrow 13\}$, $v^{*}=3$, and $\mathfrak{M}={\langle 2187,253\rangle }^4$; and  super-singular   cases $c=\mathrm{69,979}$ with symbol $\{769\leftrightarrow 7\leftarrow 13\}$, $v^{*}=4$, and $\mathfrak{M}={\langle 6561,676|677\rangle }^4$ of elevated coclass 3; and $c=\mathrm{86,821}$ with symbol $\{79\leftrightarrow 157\leftarrow 7\}$, $v^{*}=4$, and $\mathfrak{M}={\langle 729,37..39\rangle }^4$.

In

Table 17. Prototypes for Graph III.7.

No.	c	$\mathit{q}\leftrightarrow \mathit{p}\leftarrow \mathit{r}$	${\mathit{v}}^{*}$	v	$\mathit{m},\mathit{n}$	$\tilde{\mathit{v}}$	$\tilde{\mathit{m}},\tilde{\mathit{n}}$	Capitulation Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	4599	$9\leftrightarrow 73\leftarrow 7$	1	2	$1,2$	2	$1,2$	$\mathrm{a}.3^{*}$	$\langle 81,7\rangle$	$=2$
2	$\mathrm{12,051}$	$13\leftrightarrow 103\leftarrow 9$	1	2	$1,1$	2	$1,1$	$\mathrm{a}.3^{*}$	$\langle 81,7\rangle$	$=2$
6	$\mathrm{31,707}$	$9\leftrightarrow 271\leftarrow 13$	2	2	$1,0$	2	$1,0$	$\mathrm{a}.3^{*}$	$\langle 81,7\rangle$	$=2$
21	$\mathrm{76,741}$	$577\leftrightarrow 19\leftarrow 7$	2	2	$0,1$	2	$0,1$	$\mathrm{H}.4$	$\langle 2187,65|67\rangle$	$\ge 3$
27	$\mathrm{90,243}$	$271\leftrightarrow 9\leftarrow 37$	2	2	$0,1$	2	$0,1$	$\mathrm{d}.19$	$\langle 729,41\rangle$	$\ge 2$
13	$\mathrm{61,243}$	$673\leftrightarrow 7\leftarrow 13$	3	3	$0,1$	3	$1,0$	$\mathrm{b}.10$	$\langle 2187,253\rangle$	$\ge 2$
17	$\mathrm{69,979}$	$769\leftrightarrow 7\leftarrow 13$	4	3	$0,1$	3	$0,1$	$\mathrm{d}.19$	$\langle 6561,676|$677〉	$\ge 3$
25	$\mathrm{86,821}$	$79\leftrightarrow 157\leftarrow 7$	4	3	$1,1$	3	$1,2$	$\mathrm{b}.10$	$\langle 729,37..39\rangle$	$\ge 2$

, we summarize the prototypes of Graph $\mathrm{III}.7$ in the same way as in Table 13.

8.4. Category III, Graph 8

Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields belonging to Graph 8 of Category $\mathrm{III}$ with the combined cubic residue symbol ${[p,q,r]}_3=\{r\to p\leftrightarrow q\leftarrow r\}$ of three prime(power)s dividing the conductor $c=pqr$.

Proposition 12

(Quartet with 3-rank two for $\mathrm{III}.8$). For fixed $\mu \in \{1,2,3,4\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$; then, the principal factor of $k_{\mu }$ is $A\left(k_{\mu }\right)=r$, and the 3-class group of $k_{\mu }$ is

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{p}\right],\left[\mathfrak{q}\right]\rangle \simeq (3,3).$$

(68)

The unramified cyclic cubic relative extensions of $k_{\mu }$ are among the absolutely bicyclic bicubic fields $B_i$, $1\le i\le 10$. The  wild   ranks for $i=1,2,3,4,7,10$ are $r_i=2$, independently of $m,n,\tilde{m},\tilde{n}$. For each μ, there are two tame extensions $B_j/k_{\mu }$, $B_{\mathsf{\ell }}/k_{\mu }$ with the following properties.

• The first, $B_j$, has the norm class group $N_{B_j/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_j\right)\right)=\langle \left[\mathfrak{p}\right]\rangle$, the  cyclic   transfer kernel

  $$ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$$

  (69)

  of order 3, and the  elementary tricyclic  3-class group ${\mathrm{Cl}}_3\left(B_j\right)=\langle \left[{\mathfrak{P}}_1\right],\left[{\mathfrak{P}}_2\right],\left[{\mathfrak{P}}_3\right]\rangle \simeq (3,3,3)$, generated by the classes of the prime ideals of $B_j$ over $\mathfrak{p}{\mathcal{O}}_{B_j}={\mathfrak{P}}_1{\mathfrak{P}}_2{\mathfrak{P}}_3$.
• The second, $B_{\mathsf{\ell }}$, has the norm class group $N_{B_{\mathsf{\ell }}/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$,  cyclic   transfer kernel

  $$ker\left(T_{B_{\mathsf{\ell }}/k_{\mu }}\right)=\langle \left[\mathfrak{p}\right]\rangle$$

  (70)

  of order 3, and  elementary tricyclic  3-class group ${\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)=\langle \left[{\mathfrak{Q}}_1\right],\left[{\mathfrak{Q}}_2\right],\left[{\mathfrak{Q}}_3\right]\rangle \simeq (3,3,3)$, generated by the classes of the prime ideals of $B_{\mathsf{\ell }}$ over $\mathfrak{q}{\mathcal{O}}_{B_{\mathsf{\ell }}}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$.
• The pair $(j,\mathsf{\ell })$ forms a  mandatory transposition   of the transfer kernel type $\varkappa \left(k_{\mu }\right)$. The remaining two $B_i>k_{\mu }$, $i\ne j$, $i\ne \mathsf{\ell }$, have the norm class group $\langle \left[\mathfrak{pq}\right]\rangle$ or $\langle \left[{\mathfrak{pq}}^2\right]\rangle$, necessarily a  non-elementary bicyclic   3-class group of order $27\mid h_3\left(B_i\right)=\frac{(U_i:V_i)}{3}{h}_3\left(k_{pq}\right)$ and $\frac{(U_i:V_i)}{3}{h}_3\left({\tilde{k}}_{pq}\right)$, respectively, and the transfer kernels

  $$ker\left(T_{B_i/k_{\mu }}\right)\ge \langle \left[{\mathfrak{p}}^m{\mathfrak{q}}^n\right]\rangle ,\mathit{and}\ge \langle \left[{\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}\right]\rangle ,$$

  respectively, providing the option of a possible  fixed point   in the transfer kernel type $\varkappa \left(k_{\mu }\right)$.

Proof. 

Since $r\to p$, there are two principal factors $A\left(k_{pr}\right)=A\left({\tilde{k}}_{pr}\right)=r$. Since $q\leftarrow r$, there are two further principal factors $A\left(k_{qr}\right)=A\left({\tilde{k}}_{qr}\right)=r$. Since $q\leftarrow r\to p$ is universally repelling, we also have four other principal factors $A\left(k_{\mu }\right)=r$, for all $1\le \mu \le 4$ according to [2] (Prop. 4.6, p. 49). Since $\mathfrak{r}=\alpha {\mathcal{O}}_{k_{\mu }}$ is a principal ideal, its class $\left[\mathfrak{r}\right]=1$ is trivial, whereas the classes $\left[\mathfrak{p}\right],\left[\mathfrak{q}\right]$ are non-trivial and generate ${\mathrm{Cl}}_3\left(k_{\mu }\right)$.

There are four tame bicyclic bicubic fields, $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$, $B_8=k_2{k}_4{k}_p{k}_{qr}$, $B_9=k_2{k}_3{k}_q{k}_{pr}$, satisfying $9\mid h_3\left(B_i\right)=(U_i:V_i)$, for $i\in \{5,6,8,9\}$. Consequently, we must have the indices $I_i=(U_i:V_i)\in \{9,27\}$, and thus the matrix ranks $r_5=r_6=r_8=r_9=2$.

On the other hand, there are six wild bicyclic bicubic fields, $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, with $h_3\left(B_i\right)>(U_i:V_i)$.

For these bicyclic bicubic fields $B_i$, $i\in \{1,2,3,4,7,10\}$, the rank $r_i$ is calculated with row operations on the associated principal factor matrices $M_i$:

$M_1=M_2=\left(\begin{array}{ccc}0& 0& 1\\ m& n& 0\\ 0& 0& 1\\ 0& 0& 1\end{array}\right)$, $M_{10}=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 0& 0& 1\\ m& n& 0\end{array}\right)$, $M_3=M_4=\left(\begin{array}{ccc}0& 0& 1\\ \tilde{m}& \tilde{n}& 0\\ 0& 0& 1\\ 0& 0& 1\end{array}\right)$, $M_7=\left(\begin{array}{ccc}0& 0& 1\\ 0& 0& 1\\ 0& 0& 1\\ \tilde{m}& \tilde{n}& 0\end{array}\right)$.

For $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$ and $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $M_1=M_2$ immediately leads to rank $r_1=r_2=2$, since $(m,n)\ne (0,0)$, and similarly, for $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $M_{10}$ leads to rank $r_{10}=2$.

For $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, and $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_3=M_4$ immediately leads to rank $r_3=r_4=2$, since $(\tilde{m},\tilde{n})\ne (0,0)$, and similarly, for $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ leads to rank $r_7=2$.

• So, Graph 8 of Category III is the unique situation where $r_i=2$, for all $1\le i\le 10$, without any conditions, and thus, $h_3\left(B_i\right)=\frac{I_i}{3}{h}_3\left(k_{pq}\right)$, for $i\in \{1,2,10\}$, and $h_3\left(B_i\right)=\frac{I_i}{3}{h}_3\left({\tilde{k}}_{pq}\right)$, for $i\in \{3,4,7\}$, where $I_i=(U_i:V_i)\in \{9,27\}$, and $9\mid h_3\left(k_{pq}\right)$, $9\mid h_3\left({\tilde{k}}_{pq}\right)$.

In each case, the minimal subfield unit index $(U_i:V_i)=9$ corresponds to the maximal unit norm index $(U\left(k_{\mu }\right):N_{B_i/k_{\mu }}\left(U_i\right))=3$, associated with a total transfer kernel $\#ker\left(T_{B_i/k_{\mu }}\right)=9$, whenever $k_{\mu }<B_i$$1\le \mu \le 4$, $1\le i\le 10$.

According to Theorem 8, the unramified cyclic cubic relative extensions of $k_{\mu }$ among the absolutely bicyclic bicubic subfields of the 3-genus field $k^{*}=k_p{k}_q{k}_r$ are $B_1,B_5,B_6,B_7$, for $\mu =1$, $B_2,B_7,B_8,B_9$, for $\mu =2$, $B_3,B_5,B_9,B_{10}$, for $\mu =3$, and $B_4,B_6,B_8,B_{10}$, for $\mu =4$.

Since p splits in $k_q$, it also splits in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$.

Since $\mathfrak{q}$ is principal in $k_q$, $\left[\mathfrak{q}\right]$ capitulates in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$.

For $(\mu ,j)\in \left\{\right(1,6),(4,6),(2,9),(3,9\left)\right\}$, the minimal unit norm index $(U\left(k_{\mu }\right):N_{B_j/k_{\mu }}\left(U_j\right))=1$, associated to the partial transfer kernel $ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$, corresponds to the maximal subfield unit index $h_3\left(B_j\right)=(U_j:V_j)=27$, giving rise to the characteristic abelian-type invariants ${\mathrm{Cl}}_3\left(B_j\right)=\langle \left[{\mathfrak{P}}_1\right],\left[{\mathfrak{P}}_2\right],\left[{\mathfrak{P}}_3\right]\rangle \simeq (3,3,3)$ generated by the classes of the prime ideals of $B_j$ over $\mathfrak{p}{\mathcal{O}}_{B_j}={\mathfrak{P}}_1{\mathfrak{P}}_2{\mathfrak{P}}_3$. The field $B_j$, which contains $k_{\mu }$, has the norm class group $N_{B_j/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_j\right)\right)=\langle \left[\mathfrak{p}\right]\rangle$.

Since q splits in $k_p$, it also splits in $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$ and $B_8=k_2{k}_4{k}_p{k}_{qr}$.

Since $\mathfrak{p}$ is principal in $k_p$, $\left[\mathfrak{p}\right]$ capitulates in $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$ and $B_8=k_2{k}_4{k}_p{k}_{qr}$.

For $(\mu ,\mathsf{\ell })\in \left\{\right(1,5),(3,5),(2,8),(4,8\left)\right\}$, the minimal unit norm index $(U\left(k_{\mu }\right):N_{B_{\mathsf{\ell }}/k_{\mu }}\left(U_{\mathsf{\ell }}\right))=1$, associated to the partial transfer kernel $ker\left(T_{B_{\mathsf{\ell }}/k_{\mu }}\right)=\langle \left[\mathfrak{p}\right]\rangle$, corresponds to the maximal subfield unit index $h_3\left(B_{\mathsf{\ell }}\right)=(U_{\mathsf{\ell }}:V_{\mathsf{\ell }})=27$, giving rise to the characteristic abelian-type invariants ${\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)=\langle \left[{\mathfrak{Q}}_1\right],\left[{\mathfrak{Q}}_2\right],\left[{\mathfrak{Q}}_3\right]\rangle \simeq (3,3,3)$ generated by the classes of the prime ideals of $B_{\mathsf{\ell }}$ over $\mathfrak{q}{\mathcal{O}}_{B_{\mathsf{\ell }}}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$. The field $B_{\mathsf{\ell }}$, which contains $k_{\mu }$, has the norm class group $N_{B_{\mathsf{\ell }}/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)\right)=\langle \left[\mathfrak{q}\right]\rangle$.

Since ${\mathfrak{p}}^m{\mathfrak{q}}^n$ is principal in $k_{pq}$, $\left[{\mathfrak{p}}^m{\mathfrak{q}}^n\right]$ capitulates in $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, and $B_{10}=k_3{k}_4{k}_r{k}_{pq}$.

Since ${\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}$ is principal in ${\tilde{k}}_{pq}$, $\left[{\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}\right]$ capitulates in $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, and $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$.

The remaining two $B_i>k_{\mu }$, $i\in \{1,2,3,4,7,10\}$, more precisely, $i\in \{1,7\}$ for $\mu =1$, and $i\in \{2,7\}$ for $\mu =2$, and $i\in \{3,10\}$ for $\mu =3$, and $i\in \{4,10\}$ for $\mu =4$, have the norm class group $\langle \left[\mathfrak{pq}\right]\rangle$ or $\langle \left[{\mathfrak{pq}}^2\right]\rangle$, and the minimal transfer kernels $ker\left(T_{B_i/k_{\mu }}\right)\ge \langle \left[{\mathfrak{p}}^m{\mathfrak{q}}^n\right]\rangle$ and $ker\left(T_{B_i/k_{\mu }}\right)\ge \langle \left[{\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}\right]\rangle$, respectively. □

Proposition 12 and parts of its proof are now summarized in

Table 18. Norm class groups and minimal transfer kernels for Graph III.8.

Base	${\mathit{k}}_1$				${\mathit{k}}_2$				${\mathit{k}}_3$				${\mathit{k}}_4$
Ext	${\mathit{B}}_1$	${\mathit{B}}_5$	${\mathit{B}}_6$	${\mathit{B}}_7$	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$	${\mathit{B}}_4$	${\mathit{B}}_6$	${\mathit{B}}_8$	${\mathit{B}}_{10}$
NCG	$\mathfrak{pq}$	$\mathfrak{q}$	$\mathfrak{p}$	${\mathfrak{pq}}^2$	$\mathfrak{pq}$	${\mathfrak{pq}}^2$	$\mathfrak{q}$	$\mathfrak{p}$	$\mathfrak{pq}$	$\mathfrak{q}$	$\mathfrak{p}$	${\mathfrak{pq}}^2$	$\mathfrak{pq}$	$\mathfrak{p}$	$\mathfrak{q}$	${\mathfrak{pq}}^2$
TK	${\mathfrak{p}}^m{\mathfrak{q}}^n$	$\mathfrak{p}$	$\mathfrak{q}$	${\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}$	${\mathfrak{p}}^m{\mathfrak{q}}^n$	${\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}$	$\mathfrak{p}$	$\mathfrak{q}$	${\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}$	$\mathfrak{p}$	$\mathfrak{q}$	${\mathfrak{p}}^m{\mathfrak{q}}^n$	${\mathfrak{p}}^{\tilde{m}}{\mathfrak{q}}^{\tilde{n}}$	$\mathfrak{q}$	$\mathfrak{p}$	${\mathfrak{p}}^m{\mathfrak{q}}^n$
$\varkappa$	x	$\mathbf{3}$	$\mathbf{2}$	$\tilde{x}$	y	$\tilde{y}$	$\mathbf{4}$	$\mathbf{3}$	$\tilde{z}$	$\mathbf{3}$	$\mathbf{2}$	z	$\tilde{w}$	$\mathbf{3}$	$\mathbf{2}$	w

, with transposition in boldface font, based on Corollary 2. In this table, we give the norm class group (NCG) $N_{B_i/k_{\mu }}\left({\mathrm{Cl}}_3\left(B_i\right)\right)$ and the transfer kernel (TK) $ker\left(T_{B_i/k_{\mu }}\right)$, also in the symbolic form $\varkappa$ with place holders $1\le x,\tilde{x},y,\tilde{y},z,\tilde{z},w,\tilde{w}\le 4$, for each collection of four unramified cyclic cubic relative extensions $B_j$, $j=1,\dots ,10$, of each base field $k_{\mu }$, $\mu =1,\dots ,4$, of the quartet.

Theorem 18

(Second 3-class group for $\mathrm{III}.8$). Let $(k_1,\dots ,k_4)$ be the quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 8 of Category $\mathrm{III}$ with combined cubic residue symbol ${[p,q,r]}_3=\{r\to p\leftrightarrow q\leftarrow r\}$.

To identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $1\le \mu \le 4$, let the  principal factor   of $k_{pq}$ and ${\tilde{k}}_{pq}$, respectively, be $A\left(k_{pq}\right)=p^m{q}^n$ and $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, respectively, and additionally, assume the  regular  situation where both ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq {\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)\simeq (3,3)$ are elementary bicyclic, whence $(m,n)=(\tilde{m},\tilde{n})$.

Then, there are several  minimal transfer kernel types   (mTKT) ${\varkappa }_0$ of $k_{\mu }$, $1\le \mu \le 4$, and the other possible capitulation types in ascending order ${\varkappa }_0<{\varkappa }^{\prime }<{\varkappa }^{″}$, ending in the mandatory ${\varkappa }^{″}=\left(2100\right)$, type $\mathrm{b}.10$:

• either ${\varkappa }_0=\left(2111\right)$, type $\mathrm{H}.4$, ${\varkappa }^{\prime }=\left(2110\right)$, type $\mathrm{d}.19$, for $\mathcal{P}=1$, or ${\varkappa }_0=\left(2133\right)$, type $\mathrm{F}.11$, ${\varkappa }^{\prime }=\left(2130\right)$, type $\mathrm{d}.23$, or $\left(2103\right)$, type $\mathrm{d}.25$, for $\mathcal{P}=2$, and the second 3-class group is $\mathfrak{M}\simeq$

  $$\left\{\begin{array}{cc}\langle 729,34..39\rangle ,\alpha =[111,111,21,21],\varkappa =\left(2100\right)\hfill & \mathit{if}\mathcal{N}=2,\hfill \\ \langle 729,41\rangle ,\alpha =[111,111,22,21],\varkappa =\left(2110\right)\hfill & \mathit{if}\mathcal{P}=1,\mathcal{N}=3,\hfill \\ \langle 729,42|43\rangle ,\alpha =[111,111,22,21],\varkappa =(2130\left)\right|\left(2103\right)\hfill & \mathit{if}\mathcal{P}=2,\mathcal{N}=3,\hfill \\ \langle 2187,65|67\rangle ,\alpha =[111,111,22,22],\varkappa =(2111)\hfill & \mathit{if}\mathcal{P}=1,\mathcal{N}=4,\hfill \\ \langle 2187,66|73\rangle ,\alpha =[111,111,22,22],\varkappa =(2133)\hfill & \mathit{if}\mathcal{P}=2,\mathcal{N}=4,\hfill \end{array}\right.$$

  (71)

  where $\mathcal{N}:=\#\{1\le j\le 10\mid k_{\mu }<B_j,I_j=27\}$ and $\mathcal{P}$ is the number of prime divisors of $p^m{q}^n$. In any case, the 3-class field tower may have a group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })$ bigger than $\mathfrak{M}$ although $d_2\left(\mathfrak{M}\right)\le 4$. Further, $\mathrm{III}.8$ is the unique graph where the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$ cannot be of maximal class.

Since the group order cannot be specified in the  (super-)singular   situation, only the capitulation type can be given. Additionally, the number $\tilde{\mathcal{P}}$ of prime divisors of $p^{\tilde{m}}{q}^{\tilde{n}}$ is used, and two cases are separated.

• If $(m,n)=(\tilde{m},\tilde{n})$, then $\mathcal{P}=\tilde{\mathcal{P}}$, and all four types $\varkappa \left(k_{\mu }\right)$, $\mu =1,\dots ,4$, coincide:

  $$\left\{\begin{array}{cc}\varkappa =\left(2100\right),\mathrm{b}.10\hfill & \mathit{if}\mathcal{N}=2,\hfill \\ \varkappa =\left(2111\right),\mathrm{H}.4\hfill & \mathit{if}\mathcal{P}=1,\mathcal{N}=4,\hfill \\ \varkappa =\left(2133\right),\mathrm{F}.11\hfill & \mathit{if}\mathcal{P}=2,\mathcal{N}=4.\hfill \end{array}\right.$$

  (72)

  If $(m,n)\ne (\tilde{m},\tilde{n})$, then the number of coinciding types must be indicated by formal exponents:

  $$\left\{\begin{array}{cc}\varkappa ={\left(2100\right)}^4,\mathrm{b}.10\hfill & \mathit{if}\mathcal{N}=2,\hfill \\ \varkappa ={\left(2130\right)}^2,\mathrm{d}.23,\varkappa ={\left(2103\right)}^2,\mathrm{d}.25\hfill & \mathit{if}\mathcal{N}=3,\hfill \\ \varkappa ={\left(2112\right)}^4,\mathrm{F}.7\hfill & \mathit{if}\mathcal{P}=\tilde{\mathcal{P}}=1,\mathcal{N}=4,\hfill \\ \varkappa ={\left(2134\right)}^2,\mathrm{G}.16,\varkappa ={\left(2143\right)}^2,\mathrm{G}.19\hfill & \mathit{if}\mathcal{P}=\tilde{\mathcal{P}}=2,\mathcal{N}=4,\hfill \\ \varkappa ={\left(2131\right)}^2,\mathrm{F}.12,\varkappa ={\left(2113\right)}^2,\mathrm{F}.13\hfill & \mathit{if}\mathcal{P}\ne \tilde{\mathcal{P}},\mathcal{N}=4.\hfill \end{array}\right.$$

  (73)

Proof. 

We normalize the transpositions in Table 18 by the following convention: $\varkappa =(21\ast \ast )\sim (\ast 32\ast )\sim (\ast \ast 43)$, taking the leading type of three equivalent types.

Since the transfer kernels $ker\left(T_{B_i/k_{\mu }}\right)$ for the tame extensions with $i\in \{5,6,8,9\}$ are partial, the corresponding indices $I_i=(U_i:V_i)=27$ of subfield units must be maximal, whence necessarily $\mathcal{N}\ge 2$. The associated 3-class numbers $h_3\left(B_i\right)=27$ are consistent with the occurrence of two elementary tricyclic 3-class groups ${\mathrm{Cl}}_3\left(B_i\right)\simeq (3,3,3)$, connected with a transposition in the capitulation type $\varkappa \left(k_{\mu }\right)=(21\ast \ast )$, according to Proposition 12.

In the proof of this proposition, it was also derived that due to $r_i=2$, the 3-class numbers of the wild extensions are given by $h_3\left(B_i\right)=\frac{I_i}{3}{h}_3\left(k_{pq}\right)\ge 27$, for $i\in \{1,2,10\}$, and $h_3\left(B_i\right)=\frac{I_i}{3}{h}_3\left({\tilde{k}}_{pq}\right)\ge 27$, for $i\in \{3,4,7\}$, where $I_i=(U_i:V_i)\in \{9,27\}$, and $9\mid h_3\left(k_{pq}\right)$, $9\mid h_3\left({\tilde{k}}_{pq}\right)$.

It follows that the maximal class $\mathrm{cc}\left(\mathfrak{M}\right)=1$ is prohibited for two reasons: firstly, by the Artin pattern $\varkappa \sim \left(21{\varkappa }_3{\varkappa }_4\right)$, $\alpha \sim [111,111,{\alpha }_3,{\alpha }_4]$; and secondly, by the bipolarization of an order of at least 27, which implies that ${\alpha }_3$ and ${\alpha }_4$ are bicyclic equal to $\left(21\right)$ or bigger [25] (pp. 289–292).

In fact, even $\mathrm{cc}\left(\mathfrak{M}\right)=2$ is very restricted, because the candidates for $\mathfrak{M}$ must be descendants of the group $\langle 243,3\rangle$. The other two groups with two or three components $\left(111\right)$ in the abelian-type invariants $\alpha$ are discouraged, since $\varkappa \sim \left(1133\right)$, $\alpha \sim [21,111,21,111]$ for $\langle 243,7\rangle$ does not contain a transposition, and in $\varkappa \sim \left(2111\right)$, $\alpha \sim [111,21,111,111]$ for $\langle 243,4\rangle$, the transposition in $\varkappa$ is not associated with two elementary tricyclic components of $\alpha$.

If $\mathcal{N}=2$, then the Artin pattern $\alpha =[111,111,21,21]$, $\varkappa =\left(2100\right)$ identifies one of the six groups $\langle 729,34..39\rangle ,\alpha =[111,111,21,21]$, since $\langle 243,3\rangle$ is forbidden by Corollary 4.

If $\mathcal{N}=3$, then generally $\alpha =[111,111,22,21]$, with bipolarization consisting of copolarization $\left(21\right)$, i.e., coclass 2, and polarization $\left(22\right)$, i.e., class 4. Now, if $\mathcal{P}=1$, then the capitulation type $\varkappa =\left(2110\right)\sim \left(2120\right)$ contains a repetition, which identifies the group $\langle 729,41\rangle$. On the other hand, if $\mathcal{P}=2$, then the capitulation type $\varkappa =\left(2130\right)$ either contains a fixed point, which gives $\langle 729,42\rangle$, or $\varkappa =\left(2103\right)$ neither contains a repetition nor a fixed point, which gives $\langle 729,43\rangle$.

If $\mathcal{N}=4$, then generally $\alpha =[111,111,22,22]$, but a finer distinction is provided by $\mathcal{P}$. If $\mathcal{P}=1$, then the capitulation type $\varkappa =\left(2111\right)\sim \left(2122\right)$ contains two repetitions and becomes nearly constant, which identifies the groups $\langle 2187,65|67\rangle$ of coclass 3. However, if $\mathcal{P}=2$, then a fixed point and its repetition occurs in the capitulation type $\varkappa =\left(2133\right)\sim \left(2144\right)$, which leads to the groups $\langle 2187,66|73\rangle ,\alpha =[111,111,22,22]$.

Concerning the (super-)singular situation, two cases are distinguished. If $(m,n)=(\tilde{m},\tilde{n})$, then $\mathcal{P}=1$, i.e., $(m,n)\in \left\{\right(0,1),(1,0\left)\right\}$ implies two identical repetitions in ${\varkappa }_0\sim \left(2111\right)\sim \left(2122\right)$, $\mathrm{H}.4$, but $\mathcal{P}=2$, i.e., $(m,n)\in \left\{\right(1,1),(1,2),(2,1\left)\right\}$, produces a single fixed point in ${\varkappa }_0\sim \left(2133\right)\sim \left(2144\right)$, $\mathrm{F}.11$. These two minimal transfer kernel types for $\mathcal{N}=4$ both expand to ${\varkappa }^{″}\sim \left(2100\right)$ for $\mathcal{N}=2$. All three cases are uniform. If $\mathcal{N}=3$ were possible, then $\mathcal{P}=1$ would lead to type $\varkappa \sim \left(2110\right)\sim \left(2120\right)$, $\mathrm{d}.19$, and $\mathcal{P}=2$ would either imply type $\varkappa \sim \left(2130\right)$, $\mathrm{d}.23$, or $\varkappa \sim \left(2103\right)$, $\mathrm{d}.25$. The latter case would be non-uniform, but $\mathcal{N}=3$ does not seem to occur at all.

If $(m,n)\ne (\tilde{m},\tilde{n})$, then nevertheless, $\mathcal{P}=\tilde{\mathcal{P}}$ is possible, and then $\mathcal{P}=1$ implies two distinct repetitions in $\varkappa \sim \left(2112\right)\sim \left(2121\right)$, $\mathrm{F}.7$, uniformly, whereas $\mathcal{P}=2$ leads to either two fixed points in $\varkappa \sim \left(2134\right)$, $\mathrm{G}.16$ or a second transposition in $\varkappa \sim \left(2143\right)$, $\mathrm{G}.19$. These permutation types would be non-uniform in two sub-doublets, but they are obviously forbidden, for an unknown reason. Finally, $\mathcal{P}\ne \tilde{\mathcal{P}}$ admits several distinct realizations with identical result: it always leads to a repetition, and additionally either to a fixed point in $\varkappa \sim \left(2131\right)$, $\mathrm{F}.12$, or a non-fixed point in $\varkappa \sim \left(2113\right)$, $\mathrm{F}.13$, non-uniformly in two sub-doublets. □

Corollary 12

(Non-uniformity of the quartet for $\mathrm{III}.8$). If $(m,n)=(\tilde{m},\tilde{n})$, in particular always in the regular situation, the components of the quartet, all with 3-rank two, uniformly share a common capitulation type $\varkappa \left(k_{\mu }\right)$, common abelian-type invariants $\alpha \left(k_{\mu }\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, for $1\le \mu \le 4$. Otherwise, the invariants may be non-uniform, divided in two sub-doublets.

Proof. 

In the regular case, we must have $(m,n)=(\tilde{m},\tilde{n})$. All TKTs are either equivalent to F.11, with a mandatory fixed point, if ${\mathfrak{p}}^m{\mathfrak{q}}^n\in \{\mathfrak{p}\mathfrak{q},\mathfrak{p}{\mathfrak{q}}^2\}$, or to H.4, if ${\mathfrak{p}}^m{\mathfrak{q}}^n\in \{\mathfrak{p},\mathfrak{q}\}$, according to Table 18. The potential non-uniformity was proved in Theorem 18. □

In

Table 19. Prototypes for Graph III.8.

No.	c	$\mathit{r}\to \mathit{p}\leftrightarrow \mathit{q}\leftarrow \mathit{r}$	${\mathit{v}}^{*}$	v	$\mathit{m},\mathit{n}$	$\tilde{\mathit{v}}$	$\tilde{\mathit{m}},\tilde{\mathit{n}}$	Capitulation Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	$20293$	$13\to 7\leftrightarrow 223\leftarrow 13$	1	2	$2,1$			$\mathrm{b}.10$	$\langle 729,37..39\rangle$	$\ge 2$
2	$41509$	$31\to 13\leftrightarrow 103\leftarrow 31$	1	2	$1,1$			$\mathrm{b}.10$	$\langle 729,37..39\rangle$	$\ge 2$
3	$46341$	$19\to 9\leftrightarrow 271\leftarrow 19$	2	2	$0,1$			$\mathrm{b}.10$	$\langle 729,37..39\rangle$	$\ge 2$
5	$52497$	$19\to 9\leftrightarrow 307\leftarrow 19$	1	2	$2,1$			$\mathrm{F}.11$	$\langle 2187,66|73\rangle$	$\ge 2$
7	$92911$	$13\to 7\leftrightarrow 1021\leftarrow 13$	1	2	$1,1$			$\mathrm{F}.11$	$\langle 2187,66|73\rangle$	$\ge 2$
18	$191007$	$19\to 9\leftrightarrow 1117\leftarrow 19$	2	2	$1,0$			$\mathrm{b}.10$	$\langle 729,37..39\rangle$	$\ge 2$
26	$231469$	$43\to 7\leftrightarrow 769\leftarrow 43$	4	3	$0,1$	3	$0,1$	$\mathrm{H}.4$	$P_7-\#2;34|35$	$\ge 3$
40	$387729$	$9\to 67\leftrightarrow 643\leftarrow 9$	3	3	$2,1$	3	$2,1$	$\mathrm{F}.11$	$P_7-\#2;36|38$	$\ge 2$
92	$756499$	$43\to 73\leftrightarrow 241\leftarrow 43$	3	3	$0,1$	3	$1,1$	$\mathrm{F}.12$	$P_7-\#2;43\left|46\right|51|53$	$\ge 2$
								$\mathrm{F}.13$	$P_7-\#2;41\left|47\right|50|52$	$\ge 3$
93	$758233$	$7\to 19\leftrightarrow 5701\leftarrow 7$	3	3	$1,1$	3	$1,1$	$\mathrm{F}.11$	$P_7-\#2;36|38$	$\ge 2$
101	$806869$	$7\to 73\leftrightarrow 1579\leftarrow 7$	5	3	$1,1$	5	$1,1$	$\mathrm{F}.11$		$\ge 2$
105	$831001$	$67\to 79\leftrightarrow 157\leftarrow 67$	4	3	$1,1$	3	$2,1$	$\mathrm{d}.23$	$\langle 6561,678\rangle$	$\ge 3$
								$\mathrm{d}.25$	$\langle 6561,679|680\rangle$	$\ge 3$
102	$945117$	$19\to 9\leftrightarrow 5527\leftarrow 19$	3	3	$1,0$	3	$1,1$	$\mathrm{F}.12$	$P_7-\#2;43\left|46\right|51|53$	$\ge 2$
								$\mathrm{F}.13$	$P_7-\#2;41\left|47\right|50|52$	$\ge 3$
162	$1301287$	$31\to 13\leftrightarrow 3229\leftarrow 31$	4	4	$0,1$	3	$2,1$	$\mathrm{d}.23$		$\ge 2$
								$\mathrm{d}.25$		$\ge 2$
164	$1305937$	$31\to 103\leftrightarrow 409\leftarrow 31$	6	4	$1,1$	5	$1,1$	$\mathrm{F}.11$		$\ge 2$
183	$1463917$	$13\to 7\leftrightarrow 16087\leftarrow 13$	3	3	$0,1$	3	$1,0$	$\mathrm{F}.7$	$P_7-\#2;55\left|56\right|58$	$\ge 2$
185	$1483767$	$19\to 9\leftrightarrow 8677\leftarrow 19$	3	3	$2,1$	3	$0,1$	$\mathrm{F}.12$	$P_7-\#2;43\left|46\right|51|53$	$\ge 2$
								$\mathrm{F}.13$	$P_7-\#2;41\left|47\right|50|52$	$\ge 2$
253	$2068587$	$19\to 9\leftrightarrow 12097\leftarrow 19$	5	3	$1,1$	5	$0,1$	$\mathrm{F}.12$		$\ge 2$
								$\mathrm{F}.13$		$\ge 2$
385	$2991987$	$19\to 9\leftrightarrow 17497\leftarrow 19$	3	3	$1,0$	3	$2,1$	$\mathrm{F}.12$	$P_7-\#2;43\left|46\right|51|53$	$\ge 2$
								$\mathrm{F}.13$	$P_7-\#2;41\left|47\right|50|52$	$\ge 2$
468	$3556699$	$97\to 37\leftrightarrow 991\leftarrow 97$	6	5	$1,1$	3	$2,1$	$\mathrm{d}.23$		$\ge 2$
								$\mathrm{d}.25$		$\ge 2$
651	$4686019$	$109\to 13\leftrightarrow 3307\leftarrow 109$	4	4	$0,1$	3	$1,0$	$\mathrm{F}.7$	$P_7-\#2;55\left|56\right|58$	$\ge 2$

, we summarize the prototypes of Graph $\mathrm{III}.8$ in the same way as in Table 13. The group with multifurcation of order four is abbreviated by $P_7:=\langle 2187,64\rangle$. See the table and tree diagram [24] (Section 11, pp. 96–100, Tbl. 1, Fig. 5).

Example 8.

Since $\mathrm{III}.8$ is the graph with most sparse population by far, Ayadi [2] (pp. 89–90) was unable to give any examples. We found many, but not all, prototypes. These are the minimal conductors for each scenario in Theorem 18. In the  regular   case, they have been found for $\mathcal{N}\in \{2,4\}$, but not for $\mathcal{N}=3$. There are some  regular   prototypes: $c=20293$ with symbol $\{13\to 7\leftrightarrow 223\leftarrow 13\}$, $v=1$, and $\mathfrak{M}=\langle 729,37..39\rangle$; $c=46341$ with symbol $\{19\to 9\leftrightarrow 271\leftarrow 19\}$, $v=2$, and $\mathfrak{M}=\langle 729,37..39\rangle$; $c=52497$ with symbol $\{19\to 9\leftrightarrow 307\leftarrow 19\}$, $v=1$, and $\mathfrak{M}=\langle 2187,66|73\rangle$. Furthermore, there is a  super-singular   prototype $c=231469$ with symbol $\{43\to 7\leftrightarrow 769\leftarrow 43\}$, type $\mathrm{H}.4$, and $\mathfrak{M}=\langle 2187,64\rangle -\#2;i$, $i\in \{34,35\}$, with $d_2\left(\mathfrak{M}\right)=5$, outside of the library [3], not treated by Theorem 18. For 7 examples see [8] (Tbl. 6.8, p. 53).

8.5. Category III, Graph 9

Let $(k_1,\dots ,k_4)$ be a quartet of cyclic cubic number fields sharing the common conductor $c=pqr$, belonging to Graph 9 of Category $\mathrm{III}$ with combined cubic residue symbol ${[p,q,r]}_3=\{r\leftarrow p\leftrightarrow q\leftarrow r\}$.

Proposition 13.

(Quartet with 3-rank two for $\mathrm{III}.9$.)  For fixed $\mu \in \{1,2,3,4\}$, let $\mathfrak{p},\mathfrak{q},\mathfrak{r}$ be the prime ideals of $k_{\mu }$ over $p,q,r$, that is, $p{\mathcal{O}}_{k_{\mu }}={\mathfrak{p}}^3$, $q{\mathcal{O}}_{k_{\mu }}={\mathfrak{q}}^3$, $r{\mathcal{O}}_{k_{\mu }}={\mathfrak{r}}^3$, then the  principal factor   of $k_{\mu }$ is $A\left(k_{\mu }\right)=p$, and the 3-class group of $k_{\mu }$ is

$${\mathrm{Cl}}_3\left(k_{\mu }\right)=\langle \left[\mathfrak{q}\right],\left[\mathfrak{r}\right]\rangle \simeq (3,3).$$

(74)

In terms of n and $\tilde{n}$ in $A\left(k_{pq}\right)=p^m{q}^n$ and $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, the ranks of the  wild   extensions are

$$r_1=r_2=r_{10}=3\mathit{iff}n\ne 0\mathit{iff}q\mid A\left(k_{pq}\right)\mathit{and}{r}_3=r_4=r_7=3\mathit{iff}\tilde{n}\ne 0\mathit{iff}q\mid A\left({\tilde{k}}_{pq}\right).$$

(75)

Proof. 

By Proposition 3, the principal factors are $A\left(k_{pr}\right)=A\left({\tilde{k}}_{pr}\right)=p$, since $r\leftarrow p$; and $A\left(k_{qr}\right)=A\left({\tilde{k}}_{qr}\right)=r$, since $q\leftarrow r$. Further, by Proposition 4, $A\left(k_{\mu }\right)=p$, for all $1\le \mu \le 4$, since p is universally repelling $r\leftarrow p\to q$. Since $\mathfrak{p}=\alpha {\mathcal{O}}_{k_{\mu }}$ is a principal ideal, its class $\left[\mathfrak{p}\right]=1$ is trivial, whereas the classes $\left[\mathfrak{q}\right],\left[\mathfrak{r}\right]$ are non-trivial. By Corollary 3,

• since $\mathfrak{q}$ is principal ideal in $k_q$, the class $\left[\mathfrak{q}\right]$ capitulates in $B_6=k_1{k}_4{k}_q{\tilde{k}}_{pr}$ and $B_9=k_2{k}_3{k}_q{k}_{pr}$;
• since $\mathfrak{r}$ is principal ideal in $k_r$, the class $\left[\mathfrak{r}\right]$ capitulates in $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$ and $B_{10}=k_3{k}_4{k}_r{k}_{pq}$.
• However, since $\mathfrak{r}$ is principal ideal in $k_{qr}$ and ${\tilde{k}}_{qr}$, the class $\left[\mathfrak{r}\right]$ also capitulates in $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$, $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$, $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $B_5=k_1{k}_3{k}_p{\tilde{k}}_{qr}$, and $B_8=k_2{k}_4{k}_p{k}_{qr}$.

For the wild bicyclic bicubic fields $B_j$, $j\in \{1,2,3,4,7,10\}$, the rank $r_j$ is calculated with row operations on the associated principal factor matrices $M_j$:

$M_1=M_2=\left(\begin{array}{ccc}1& 0& 0\\ m& n& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$, $M_{10}=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\\ m& n& 0\end{array}\right)$, $M_3=M_4=\left(\begin{array}{ccc}1& 0& 0\\ \tilde{m}& \tilde{n}& 0\\ 1& 0& 0\\ 0& 0& 1\end{array}\right)$, $M_7=\left(\begin{array}{ccc}1& 0& 0\\ 1& 0& 0\\ 0& 0& 1\\ \tilde{m}& \tilde{n}& 0\end{array}\right)$.

For $B_1=k_1{k}_{pq}{k}_{pr}{k}_{qr}$ and $B_2=k_2{k}_{pq}{\tilde{k}}_{pr}{\tilde{k}}_{qr}$, $M_1=M_2$ leads to the decisive pivot element n in the middle column, for $B_{10}=k_3{k}_4{k}_r{k}_{pq}$, $M_{10}$ also leads to n. So, rank $r_1=r_2=r_{10}=3$ iff $n\ne 0$.

For $B_3=k_3{\tilde{k}}_{pq}{\tilde{k}}_{pr}{k}_{qr}$ and $B_4=k_4{\tilde{k}}_{pq}{k}_{pr}{\tilde{k}}_{qr}$, $M_3=M_4$ leads to the decisive pivot element $\tilde{n}$ in the middle column; for $B_7=k_1{k}_2{k}_r{\tilde{k}}_{pq}$, $M_7$ also leads to $\tilde{n}$. So, rank $r_3=r_4=r_7=3$ iff $\tilde{n}\ne 0$. □

In terms of capitulation targets in Corollary 2, Proposition 13 and parts of its proof are now summarized in

Table 20. Norm class groups and minimal transfer kernels for Graph III.9.

Base	${\mathit{k}}_1$				${\mathit{k}}_2$				${\mathit{k}}_3$				${\mathit{k}}_4$
Ext	${\mathit{B}}_1$	${\mathit{B}}_5$	${\mathit{B}}_6$	${\mathit{B}}_7$	${\mathit{B}}_2$	${\mathit{B}}_7$	${\mathit{B}}_8$	${\mathit{B}}_9$	${\mathit{B}}_3$	${\mathit{B}}_5$	${\mathit{B}}_9$	${\mathit{B}}_{10}$	${\mathit{B}}_4$	${\mathit{B}}_6$	${\mathit{B}}_8$	${\mathit{B}}_{10}$
NCG	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	${\mathfrak{qr}}^2$	${\mathfrak{qr}}^2$	$\mathfrak{qr}$	$\mathfrak{q}$	$\mathfrak{r}$	${\mathfrak{qr}}^2$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{qr}$	$\mathfrak{qr}$	$\mathfrak{r}$	$\mathfrak{q}$	${\mathfrak{qr}}^2$
TK	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{r}$	$\mathfrak{q}$	$\mathfrak{r}$	$\mathfrak{r}$
$\varkappa$	3	$\mathbf{3}$	$\mathbf{2}$	3	4	4	$\mathbf{4}$	$\mathbf{3}$	3	$\mathbf{3}$	$\mathbf{2}$	3	2	$\mathbf{3}$	$\mathbf{2}$	2

, with transpositions in bold font.

Theorem 19

(Second 3-class group for $\mathrm{III}.9$). To identify the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, $1\le \mu \le 4$, let the principal factor of $k_{pq}$ and ${\tilde{k}}_{pq}$, respectively, be $A\left(k_{pq}\right)=p^m{q}^n$ and $A\left({\tilde{k}}_{pq}\right)=p^{\tilde{m}}{q}^{\tilde{n}}$, respectively, and additionally, assume the regular situation where both ${\mathrm{Cl}}_3\left(k_{pq}\right)\simeq {\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)\simeq (3,3)$ are elementary bicyclic, whence $(m,n)=(\tilde{m},\tilde{n})$.

Then, the  minimal transfer kernel type   (mTKT) ${\varkappa }_0$ of $k_{\mu }$, $1\le \mu \le 4$, and other possible capitulation types in ascending order ${\varkappa }_0<{\varkappa }^{\prime }<{\varkappa }^{″}<{\varkappa }^{‴}$, ending in the mandatory ${\varkappa }^{‴}=\left(2000\right)$, type $\mathrm{a}.3^{*}$, are ${\varkappa }_0=\left(2111\right)$, type $\mathrm{H}.4$, ${\varkappa }^{\prime }=\left(2110\right)$, type $\mathrm{d}.19$, ${\varkappa }^{″}=\left(2100\right)$, type $\mathrm{b}.10$, and the second 3-class group is $\mathfrak{M}\simeq$

$$\left\{\begin{array}{cc}\langle 81,7\rangle ,\alpha =[111,11,11,11],\varkappa =\left(2000\right)\hfill & \mathit{if}n\ne 0,\mathcal{N}=1,\hfill \\ \langle 729,34..39\rangle ,\alpha =[111,111,21,21],\varkappa =\left(2100\right)\hfill & \mathit{if}n=0,\mathcal{N}=2,\hfill \\ \langle 729,41\rangle ,\alpha =[111,111,22,21],\varkappa =\left(2110\right)\hfill & \mathit{if}n=0,\mathcal{N}=3,\hfill \\ \langle 2187,65|67\rangle ,\alpha =[111,111,22,22],\varkappa =(2111)\hfill & \mathit{if}n=0,\mathcal{N}=4,\hfill \end{array}\right.$$

(76)

where $\mathcal{N}:=\#\{1\le j\le 10\mid k_{\mu }<B_j,I_j=27\}$. Only in the first case does the 3-class field tower certainly have the group $\mathfrak{G}=\mathrm{Gal}({\mathrm{F}}_3^{\infty }\left(k_{\mu }\right)/k_{\mu })\simeq \mathfrak{M}$ and length ${\mathsf{\ell }}_3\left(k_{\mu }\right)=2$, otherwise ${\mathsf{\ell }}_3\left(k_{\mu }\right)\ge 3$ cannot be excluded, even if $d_2\left(\mathfrak{M}\right)\le 4$.

Proof. 

The essence of the proof is a systematic evaluation of the facts proved in Proposition 13 and illustrated by Table 20, ordered by increasing indices $I_j:=(U_j:V_j)$ of subfield units and, accordingly, by Lemma 2, shrinking transfer kernels $ker\left(T_{B_j/k_{\mu }}\right)$, with $1\le j\le 10$, $1\le \mu \le 4$.

1.

For the maximal TKT, ${\varkappa }^{‴}\sim \left(2000\right)$, called $\mathrm{a}.3^{*}$ in conjunction with ATI $[111,11,11,11]$, we must have $n\ne 0$, $\tilde{n}\ne 0$ and by (75) wild ranks $r_j=3$ and indices $I_j=3$ for all $j=1,2,3,4,7,10$, causing 8 (because 7 is used twice over $k_1$ and $k_2$ and 10 is used twice over $k_3$ and $k_4$) minimal 3-class numbers $h_3\left(B_j\right)=h_3\left(k_{pq}\right)=h_3\left({\tilde{k}}_{pq}\right)=9$, by (50), and ATI ${\mathrm{Cl}}_3\left(B_j\right)\simeq \left(11\right)$, characteristic for a group of coclass $\mathrm{cc}\left(\mathfrak{M}\right)=1$, i.e., maximal class, namely $\mathfrak{M}\simeq \langle 81,7\rangle$. However, the elementary tricyclic component $\left(111\right)$ of the ATI requires tame indices $I_j=27$ for $j=5,8$, and thus $\mathcal{N}=1$ for each $1\le \mu \le 4$ (because 5 is used twice over $k_1$ and $k_3$ and 8 is used twice over $k_2$ and $k_4$).

2.

Next, one of the total TK shrinks to a transposition, ${\varkappa }^{″}\sim \left(2100\right)$, $\mathrm{b}.10$, which requires a group of coclass $\mathrm{cc}\left(\mathfrak{M}\right)\ge 2$, implying, firstly, tame indices $I_j=27$ also for $j=6,9$, and thus $\mathcal{N}=2$ for each $1\le \mu \le 4$ (because 6 is used twice over $k_1$ and $k_4$ and 9 is used twice over $k_2$ and $k_3$), and, secondly, (from now on) necessarily both $n=\tilde{n}=0$, implying wild ranks $r_j=2$ indices $I_j\in \{9,27\}$ for all $j=1,2,3,4,7,10$, here $I_j=9$, 3-class numbers $h_3\left(B_j\right)=3·h_3\left(k_{pq}\right)=3·h_3\left({\tilde{k}}_{pq}\right)=3·9=27$, and thus ATI $\alpha \sim [111,111,21,21]$, leading to $\mathfrak{M}\simeq \langle 729,34..39\rangle$, in view of Corollary 4.

3.

Now another total TK shrinks to a repetition, ${\varkappa }^{\prime }\sim \left(2110\right)$, $\mathrm{d}.19$; the first three wild indices for $j=1,2,10$ become maximal $I_j=27$, causing $\mathcal{N}=3$ and 4 (because 10 is used twice over $k_3$ and $k_4$) maximal new 3-class numbers

$h_3\left(B_j\right)=9·h_3\left({\tilde{k}}_{pq}\right)=9·9=81$, and thus ATI $\alpha \sim [111,111,22,21]$, uniquely identifying $\mathfrak{M}\simeq \langle 729,41\rangle$.

4.

Finally, for the minimal TKT, ${\varkappa }_0\sim \left(2111\right)$, $\mathrm{H}.4$, the remaining three wild indices for $j=3,4,7$ become maximal $I_j=27$, causing $\mathcal{N}=4$ and 4 (because 7 is used twice over $k_1$ and $k_2$) maximal new 3-class numbers $h_3\left(B_j\right)=9·h_3\left({\tilde{k}}_{pq}\right)=9·9=81$, and thus ATI $\alpha \sim [111,111,22,22]$, enforcing a group of coclass $\mathrm{cc}\left(\mathfrak{M}\right)=3$ namely $\mathfrak{M}\simeq \langle 2187,65|67\rangle$.

□

Corollary 13

(Uniformity of the quartet for $\mathrm{III}.9$). The components of the quartet, all with 3-rank two, share a common capitulation type $\varkappa \left(k_{\mu }\right)$, common abelian-type invariants $\alpha \left(k_{\mu }\right)$, and a common second 3-class group $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$, for $1\le \mu \le 4$.

Proof. 

This follows immediately from Theorem 19. □

Example 9.

For 15 examples see [8] (Tbl. 6.6, p. 51). Prototypes for Graph $\mathrm{III}.9$ are the minimal conductors for each scenario in Theorem 19. They have been found for all $\mathcal{N}\in \{1,2,3,4\}$.

There are  regular   cases: $c=16471$ with symbol $\{13\leftarrow 181\leftrightarrow 7\leftarrow 13\}$, $v^{*}=1$, and $\mathfrak{G}=\mathfrak{M}=\langle 81,7\rangle$; $c=89487$ with symbol $\{9\leftarrow 163\leftrightarrow 61\leftarrow 9\}$, $v^{*}=2$, and $\mathfrak{M}=\langle 729,41\rangle$; $c=109291$ with symbol $\{7\leftarrow 13\leftrightarrow 1201\leftarrow 7\}$, $v^{*}=2$, and $\mathfrak{M}=\langle 729,34..36\rangle$; $c=193921$ with symbol $\{7\leftarrow 13\leftrightarrow 2131\leftarrow 7\}$, $v^{*}=2$, and $\mathfrak{M}=\langle 729,37..39\rangle$; and, with  extreme statistic delay, $c=707517$ with ordinal number 145, symbol $\{9\leftarrow 127\leftrightarrow 619\leftarrow 9\}$, $v^{*}=2$, and $\mathfrak{M}=\langle 2187,65|67\rangle$ with $d_2\left(\mathfrak{M}\right)=5$.

Only one  super-singular case for $c<2·{10}^5$: It is $c=197,239$ with symbol $7\leftarrow 1483\leftrightarrow 19\leftarrow 7$, $v^{*}=4$, and $\mathfrak{M}=\langle 729,37..39\rangle$. Astonishingly, there is no bigger order and coclass of $\mathfrak{M}$, due to $n\ne 0$.

In

Table 21. Prototypes for Graph III.9.

No.	c	$\mathit{r}\leftarrow \mathit{p}\leftrightarrow \mathit{q}\leftarrow \mathit{r}$	${\mathit{v}}^{*}$	v	$\mathit{m},\mathit{n}$	$\tilde{\mathit{v}}$	$\tilde{\mathit{m}},\tilde{\mathit{n}}$	Capitulation Type	$\mathfrak{M}$	${\mathit{\ell }}_3\left(\mathit{k}\right)$
1	$\mathrm{16,471}$	$13\leftarrow 181\leftrightarrow 7\leftarrow 13$	1	2	$1,1$	2	$1,1$	$\mathrm{a}.3^{*}$	$\langle 81,7\rangle$	$=2$
15	$\mathrm{89,487}$	$9\leftarrow 163\leftrightarrow 61\leftarrow 9$	2	2	$1,0$	2	$1,0$	$\mathrm{d}.19$	$\langle 729,41\rangle$	$\ge 2$
19	$\mathrm{109,291}$	$7\leftarrow 13\leftrightarrow 1201\leftarrow 7$	2	2	$1,0$	2	$1,0$	$\mathrm{b}.10$	$\langle 729,34..36\rangle$	$\ge 2$
28	$\mathrm{193,921}$	$7\leftarrow 13\leftrightarrow 2131\leftarrow 7$	2	2	$1,0$	2	$1,0$	$\mathrm{b}.10$	$\langle 729,37..39\rangle$	$\ge 2$
31	$\mathrm{197,239}$	$7\leftarrow 1483\leftrightarrow 19\leftarrow 7$	4	3	$0,1$	3	$0,1$	$\mathrm{b}.10$	$\langle 729,37..39\rangle$	$\ge 2$
145	$\mathrm{707,517}$	$9\leftarrow 127\leftrightarrow 619\leftarrow 9$	2	2	$1,0$	2	$1,0$	$\mathrm{H}.4$	$\langle 2187,65|67\rangle$	$\ge 3$

, we summarize the prototypes of Graph $\mathrm{III}.9$ in the same way as in Table 13.

9. Conclusions

In this work, we have seen that order and structure of the second 3-class group $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k\right)/k)$ of a cyclic cubic number field k with the conductor $c=pqr$ divisible by three prime(power)s $p,q,r$ and the elementary bicyclic 3-class group ${\mathrm{Cl}}_3\left(k\right)\simeq (3,3)$ depends on arithmetical invariants of other cyclic cubic auxiliary fields, associated with k. The field k is component of a quartet $(k_1,\dots ,k_4)$ of cyclic cubic fields sharing the common conductor c. The graph ${[p,q,r]}_3$, which is combined by the cubic residue symbols ${\left(\frac{p}{q}\right)}_3,{\left(\frac{q}{p}\right)}_3,{\left(\frac{p}{r}\right)}_3,{\left(\frac{r}{p}\right)}_3,{\left(\frac{q}{r}\right)}_3,{\left(\frac{r}{q}\right)}_3$ decides whether one, or two, or no component(s) of the quartet have a 3-class group of rank $\varrho \left(k_{\mu }\right)=3$, and accordingly, the conductor $c=pqr$ is called Category $\mathrm{I}$, or $\mathrm{II}$, or $\mathrm{III}$. For Category $\mathrm{I}$, the order of the 3-class group of the unique component with $\varrho \left(k_{{\mu }_0}\right)=3$ is crucial. For Category $\mathrm{II}$, the orders of both 3-class groups of the two components with $\varrho \left(k_{{\mu }_1}\right)=\varrho \left(k_{{\mu }_2}\right)=3$ exert an impact. For Category $\mathrm{III}$, the behavior is uniform with abelian $\mathfrak{M}\simeq (3,3)$, if ${[p,q,r]}_3$ does not contain mutual cubic residues (Graphs 1–4), otherwise there is exactly one pair $p\leftrightarrow q$ of mutual cubic residues (Graphs 5–9), and the auxiliary fields with decisive 3-class groups are the two subfields $k_{pq}$ and ${\tilde{k}}_{pq}$ of the absolute genus field $k^{*}$ of k, having the partial conductor $pq$. In each case, the principal factors (norms of ambiguous principal ideals) determine the fine structure in form of uniform or non-uniform second 3-class groups $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$. Explicit numerical investigations indicate that there is no upper bound for the orders of the 3-class groups ${\mathrm{Cl}}_3\left(k_{{\mu }_0}\right)$; ${\mathrm{Cl}}_3\left(k_{{\mu }_1}\right)$ and ${\mathrm{Cl}}_3\left(k_{{\mu }_2}\right)$; and ${\mathrm{Cl}}_3\left(k_{pq}\right)$ and ${\mathrm{Cl}}_3\left({\tilde{k}}_{pq}\right)$, respectively. In the regular situation, these orders are 27 and 9, respectively; in the singular situation, they are 81 and 27; but in the super-singular situation, they are at least 243 and 27, respectively, and the orders may increase unboundedly. Concrete numerical examples are known with orders up to 729.

Bicyclic bicubic fields $B_j$, $j=1,\dots ,10$, constitute the capitulation targets of the cyclic cubic fields $k_{\mu }$, $\mu =1,\dots ,4$. The introduction of important new concepts, the minimal and maximal capitulation type (mTKT), ${\varkappa }_0$ and ${\varkappa }_{\infty }$, permitted recognition of common patterns for several Graphs, partially in distinct Categories.

The four Graphs $\mathrm{II}.1$, $\mathrm{II}.2$, $\mathrm{III}.7$, $\mathrm{III}.9$ share the same ordered sequence of TKTs, ${\varkappa }_0\sim \left(2111\right)<\left(2110\right)<\left(2100\right)<\left(2000\right)\sim {\varkappa }_{\infty }$, called $\mathrm{H}.4$, $\mathrm{d}.19$, $\mathrm{b}.10$, $\mathrm{a}.3^{*}$, although the proofs and details are quite different. In terms of splitting prime ideals $\mathfrak{q}{\mathcal{O}}_{B_j}={\mathfrak{Q}}_1{\mathfrak{Q}}_2{\mathfrak{Q}}_3$, $\mathfrak{r}{\mathcal{O}}_{B_{\mathsf{\ell }}}={\mathfrak{R}}_1{\mathfrak{R}}_2{\mathfrak{R}}_3$, all these TKTs contain a crucial transposition, due to elementary tricyclic 3-class groups ${\mathrm{Cl}}_3\left(B_j\right)=\langle [{\mathfrak{Q}}_1,][{\mathfrak{Q}}_2,]\left[{\mathfrak{Q}}_3\right]\rangle$, ${\mathrm{Cl}}_3\left(B_{\mathsf{\ell }}\right)=\langle [{\mathfrak{R}}_1,][{\mathfrak{R}}_2,]\left[{\mathfrak{R}}_3\right]\rangle$, and twisted capitulation kernels $ker\left(T_{B_j/k_{\mu }}\right)=\langle \left[\mathfrak{r}\right]\rangle$, $ker\left(T_{B_{\mathsf{\ell }}/k_{\mu }}\right)=\langle \left[\mathfrak{q}\right]\rangle$, which restricts the group $\mathfrak{M}$ to descendants of $\langle 243,3\rangle$ (except $\langle 81,7\rangle$, where a total transfer kernel hides the transposition).

Similarly, the two graphs $\mathrm{I}.1$, $\mathrm{I}.2$ admit another characteristic ordered sequence of TKTs, ${\varkappa }_0\sim \left(4231\right)<\left(0231\right)<\left(0200\right),\left(0001\right)<\left(0000\right)\sim {\varkappa }_{\infty }$, called $\mathrm{G}.16$, $\mathrm{c}.21$, $\mathrm{a}.2$, $\mathrm{a}.3$, $\mathrm{a}.1$, with two fixed points, which restrict the group $\mathfrak{M}$ to descendants of $\langle 243,8\rangle$ (except $\langle 81,8\rangle$, $\langle 81,10\rangle$, $\langle 243,25\rangle$, $\langle 243,27\rangle$, where total transfer kernels partially or completely hide the fixed points).

A remarkable outsider is Graph $\mathrm{III}.8$ with a veritable wealth of exotic capitulation types, but restricted to the unusual maximal TKT ${\varkappa }_{\infty }\sim \left(2100\right)$, $\mathrm{b}.10$, forced by mandatory transposition.

Due to the lack of cubic residue conditions between the prime divisors of the conductor $c=pqr$, two Graphs $\mathrm{I}.1$, $\mathrm{III}.5$ admit the absolute maximum of all TKTs $\varkappa =\left(0000\right)$ (non-abelian!).

It might be worth one’s while to point out that a glance at ${\alpha }_2$ in Table 2 and Table 3 reveals that the commutator subgroup of all encountered second 3-class groups $\mathfrak{M}$ and 3-class tower groups $\mathfrak{G}$, respectively, has the order $\#\left({\mathfrak{M}}^{\prime }\right)\ge 9$ and $\#\left({\mathfrak{G}}^{\prime }\right)\ge 9$, respectively, which means that the class number of the Hilbert 3-class field ${\mathrm{F}}_3^1\left(k\right)$ is divisible by 9, for all cyclic cubic fields k, with the exception of $t=1$, the regular cases for $t=2$, and the Graphs $1,\dots ,4$ of Category $\mathrm{III}$ for $t=3$.

For Category $\mathrm{I}$ and $\mathrm{II}$, we expect a rather rigid impact of the groups $\mathfrak{M}=\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\mu }\right)/k_{\mu })$ for ${\mathrm{Cl}}_3\left(k_{\mu }\right)\simeq (3,3)$ on the groups $\mathrm{Gal}({\mathrm{F}}_3^2\left(k_{\nu }\right)/k_{\nu })$ for ${\mathrm{Cl}}_3\left(k_{\nu }\right)\simeq (3,3,3)$, as suggested by the numerous tables in [8]. This research line will be pursued further in a forthcoming paper.