L. V. Kuz'min, “On a family of algebraic number fields with finite 3-class field tower”, Sb. Math., 215:7 (2024), 911

Abstract: Let

$\ell=3$ ,

$k=\mathbb Q(\sqrt{-3})$ and

$K=k(\sqrt[3]{a})$ , where

$a$ is a natural number such that

$a^2\equiv 1\pmod 9$ . Under the assumption that there are exactly three places not over

$\ell$ that ramify in the extension

$K_\infty/k_\infty$ , where

$k_\infty$ and

$K_\infty$ are cyclotomic

$\mathbb Z_3$ -extensions of the fields

$k$ and

$K$ , respectively, we study 3-class field towers for intermediate fields

$K_n$ of the extension

$K_\infty/K$ .
It is shown that for each

$K_n$ the 3-class field tower of the field

$K_n$ terminates already at the first step, which means that the Galois group of the extension

$\mathbf H_\ell(K_n)/K_n$ , where

$\mathbf H_\ell(K_n)$ is the maximal unramified

$\ell$ -extension of the field

$K_n$ , is Abelian.
Bibliography: 7 titles.

§ 1. Introduction

Let

$L$ be an algebraic number field,

$\ell$ be an odd prime number, and

$\mathscr H_\ell(L)$ be the

$\ell$ -Hilbert class field of

$L$ , that is, the maximal Abelian unramified

$\ell$ -extension of the field

$L$ . It is known that the Galois group

$G(\mathscr H_\ell(L)/L)$ is canonically isomorphic to the

$\ell$ -component

$\operatorname{Cl}_\ell(L)$ of the class group of

$L$ .

Set

$F_0=L$ ,

$F_1=\mathscr H_\ell(L)$ , and for

$n>1$ let

$F_n=\mathscr H_\ell(F_{n-1})$ . The sequence of fields

$F_i$ is called the

$\ell$ -class field tower of the field

$L$ , and the field

$\mathbf H_\ell(L)=\bigcup_n F_n$ is the maximal unramified

$\ell$ -extension of

$L$ . According to the famous Golod–Shafarevich theorem, there exist fields

$L$ with an infinite

$\ell$ -class field tower, that is, fields

$L$ for which

$\mathbf H_\ell(L)$ has an infinite degree over

$L$ . Except for this fundamental fact, surprisingly little is known about

$\ell$ -class field towers. We consider the problem of finiteness of the

$\ell$ -class field tower in the following particular case.

Let

$\ell=3$ ,

$k=\mathbb Q(\sqrt{-3})$ , and let

$K=k(\sqrt[3]{a})$ be a cyclic cubic extension of the field

$k$ , where

$a\in \mathbb Z$ and

$a$ is not a cube in

$\mathbb Z$ . Thus,

$K$ is a Galois extension of the field

$\mathbb Q$ with Galois group

$S_3$ . In addition, we assume that

$K$ is Abelian over

$\ell$ , that is, the completion

$K_v$ of the field

$K$ relative to any place

$v$ over

$\ell=3$ is an Abelian extension of the field

$\mathbb Q_3$ .

Let

$k_n=\mathbb Q(\zeta_n)$ , where

$\zeta_n$ is a primitive

$\ell^{n+1}$ th root of unity (thus,

$k=k_0$ ) and

$k_\infty=\bigcup_n k_n$ is a cyclotomic

$\mathbb Z_\ell$ -extension of

$k$ . Also set

$K_n=K\cdot k_n$ , and let

$K_\infty=\bigcup_n K_n$ be a cyclotomic

$\mathbb Z_\ell$ -extension of the field

$K$ .

Theorem. Let $\ell=3$ , let $K$ have the form indicated above, and suppose that there are exactly three places not over $\ell$ that ramify the extension $K_\infty/k_\infty$ . Then $\mathbf H_\ell(K_n)=\mathscr H_\ell(K_n)$ for every $n$ , which means that the 3-class field tower of the field $K_n$ terminates already at the first step. Moreover, $G(\mathbf H_\ell(K_n)/K_n)\cong \operatorname{Cl}_\ell(K_n)$ .

In § 2 we introduce the necessary definitions and recall the notation, which is basically the same as in [1]–[4].

In § 3, using an analogue of the Riemann–Hurwitz formula we study the case where the field

$K_\infty$ has a finite Tate module (Iwasawa module). We show that for every

$n$ and any unramified

$\ell$ -extension

$L/K_n$ the Galois group

$G(L/K_n)$ acts identically on the class group

$\operatorname{Cl}_\ell(L)$ (Proposition 1). The definition of the Tate module

$T_\ell(K_\infty)$ and the related modules

$\overline T_\ell(K_\infty)$ and

$R_\ell(K_\infty)$ mentioned below is given in § 2.

In § 4, again with the use of the analogue of the Riemann–Hurwitz formula we consider the case where the module

$T_\ell(K_\infty)$ is infinite. It turns out again that in this case, for every

$n$ and any unramified Galois

$\ell$ -extension

$L/K_n$ the Galois group

$G(L/K_n)$ acts identically on the class group

$\operatorname{Cl}_\ell(L)$ (Proposition 5).

In § 5 we present a summary of the known results concerning fields

$K$ such that there are precisely three places ramifying the extension

$K_\infty/k_\infty$ and concerning the class groups

$\operatorname{Cl}_\ell(K_n)$ for such fields.

Note that in the case

$\ell=3$ the field

$K$ is in a certain sense similar to the field of rational functions on an elliptic curve. Namely, any unramified

$\ell$ -extension

$L$ of the field

$K_n$ , where

$K_n$ is an intermediate subfield of the cyclotomic

$\mathbb Z_\ell$ -extension

$K_\infty/K$ , is Abelian, similarly to the case of an elliptic curve. However, the proof of this fact is based on quite different arguments than the ones employed in the geometric situation.

§ 2. Notation and definitions

We follow basically the notation of [1]–[4]. Let

$\ell$ be a regular odd prime number and

$\zeta_n$ be a primitive

$\ell^{n+1}$ th root of unity. Set

$k=\mathbb Q(\zeta_0)$ and

$k_\infty=\bigcup_{n=1}^\infty k_n$ , where

$k_n=k(\zeta_n)$ . Let

$K=k(\sqrt[\ell]{a})$ , where

$a$ is a natural number such that a place

$v$ of the field

$k$ over

$\ell$ splits completely in the extension

$K/k$ . This means that

$a^{\ell-1}\equiv1 \pmod{\ell^2}$ . Moreover, we assume that there are exactly three places that ramify in the extension

$K_\infty/k_\infty$ . Accordingly, we assume that either

$a=p_1^{r_1}p_2^{r_2}p_3^{r_3}$ , or

$a=p^rq^s$ . In the first case

$p_1, p_2$ and

$p_3$ are prime numbers remaining prime in the extension

$k_\infty/\mathbb Q$ ; in the second case

$p$ splits in the unique quadratic subfield

$F$ of the field

$k$ into the product

$(p)=\mathfrak{p_1p_2}$ and each of the divisors

$\mathfrak p_i$ remains prime in the extension

$k_\infty/F$ , and

$\mathfrak q=(q)$ remains prime in the extension

$k_\infty/\mathbb Q$ . Accordingly, following the terminology of [1], we refer to these as extensions of type 2.1 and extensions of type 2.2. There also exist extensions of type 2.4. These are fields

$K$ of the form

$K=k(\sqrt[3]{p})$ , where

$p\equiv 8,17\pmod{27}$ .

We let

$G$ denote the Galois group

$G(K/\mathbb Q)$ ,

$H$ denote the group

$G(K/k)$ and

$\Delta$ denote the group

$G(k/\mathbb Q)$ . Thus,

$G$ is a semidirect product of

$H$ and

$\Delta$ , and the group

$\Delta$ acts on

$H$ in accordance with the Teichmüller character

$\omega\colon \Delta\to(\mathbb Z/\ell\mathbb Z)^\times$ .

We let

$\mathbb F_\ell(i)$ denote the group

$\mathbb Z/\ell\mathbb Z$ on which

$\Delta$ acts in accordance with the rule

$\omega^i$ . The index

$i$ is defined modulo

$\ell-1$ . Let

$A$ be a finite

$G$ -module which is cyclic as an

$H$ -module and satisfies

$N_H(A)=0$ , where

$N_H=\sum_{h\in H}h$ is the norm operator. Let

$A=A_0\supseteq A_1\supseteq\dots\supseteq A_n=0$ be the lower central series of the

$H$ -module

$A$ . If

$A_0/A_1\cong \mathbb F_\ell(i)$ and

$A_{n-1}\cong \mathbb F_\ell(j)$ , then we say that

$A$ starts with

$\mathbb F_\ell(i)$ and terminates with

$\mathbb F_\ell(j)$ . In this case, by Lemma 3.2 in [1] we have

$A_k/A_{k+1}{\cong}\, \mathbb F_\ell(i+k)$ for every

$k\,{<}\,n$ . If

$|A|{\kern1pt}{=}\,\ell^r$ , then

$r\equiv j-i+1\pmod{\ell-1}$ .

In this work we use an analogue of the Riemann–Hurwitz formula, which was derived by this author in [5] (a refined proof can be found in [6]). This formula, which we present in § 3, relates some linear combinations of the Iwasawa

$\lambda$ -invariants of certain Galois modules associated with a finite

$\ell$ -extension of the fields

$L'/L$ . The definition of these Galois modules is given below. Note that all fields considered in this work are

$\ell$ -extensions of the field

$k$ , and therefore all modules under study have zero Iwasawa

$\mu$ -invariants.

Let

$L$ be an arbitrary algebraic number field and

$L_\infty$ be the cyclotomic

$\mathbb Z_\ell$ -extension of

$L$ . Let

$\overline N$ denote the maximal Abelian unramified

$\ell$ -extension of the field

$L_\infty$ and

$N$ be the maximal subfield of

$\overline N$ such that all places in

$S$ (the set of all places over

$\ell$ ) split completely in

$N/L_\infty$ . We let

$T_\ell(L_\infty)$ and

$\overline T_\ell(L_\infty)$ denote the Galois groups of the extensions

$N/L_\infty$ and

$\overline N/L_\infty$ , respectively. These groups are compact Noetherian modules under the action of the group

$\Gamma=G(L_\infty/L)$ , where

$\Gamma\cong \mathbb Z_\ell$ . We fix some topological generator

$\gamma_0$ in

$\Gamma$ . Accordingly, these modules are acted upon by the Iwasawa algebra

$\Lambda=\mathbb Z_\ell[[\Gamma]]=\varprojlim\mathbb Z_\ell[\Gamma/\Gamma_n]$ , where

$\Gamma_n$ is the unique subgroup of index

$\ell^n$ in the group

$\Gamma$ . We let

$R_\ell(L_\infty)$ denote the kernel of the natural map

$\overline T_\ell(L_\infty)\to T_\ell(L_\infty)$ , which means that

$R_\ell(L_\infty)$ is the subgroup of the group

$\overline T_\ell(L_\infty)$ generated by the decomposition subgroups of all places over

$S$ .

Let

$M$ be the maximal Abelian

$\ell$ -extension of the field

$L_\infty$ that is unramified outside

$S$ , and let

$X(L_\infty)=G(M/L_\infty)$ . Then

$X(L_\infty)$ is a

$\Lambda$ -module, whose submodule of

$\Lambda$ -torsion is denoted by

$\operatorname{Tors}X(L_\infty)$ . The natural maps

$X(L_\infty)\to \overline T_\ell(L_\infty)$ and

$X(L_\infty)\!\to\! T_\ell(L_\infty)$ induce e maps

$\operatorname{Tors}X(L_\infty)\!\to\! \overline T_\ell(L_\infty)$ and

${\operatorname{Tors}X(\mkern-1mu L_\infty\mkern-1mu)\!\to\! T_\ell(\mkern-1mu L_\infty\mkern-1mu)}$ . Their images are denoted by

$\overline T'_\ell(L_\infty)$ and

$T'_\ell(L_\infty)$ , respectively. Set

$T''_\ell(L_\infty)=T_\ell(L_\infty)/T'_\ell(L_\infty)$ , and let

$\lambda'(L_\infty)$ and

$\lambda ''(L_\infty)$ denote the

$\lambda$ -invariants of the modules

$T'_\ell(L_\infty)$ and

$T''_\ell(L_\infty)$ , respectively.

We set

$R'(L_\infty)=R(L_\infty)\cap\overline T_\ell'(L_\infty)$ and

$R''(L_\infty)=R(L_\infty)/R'_\ell(L_\infty)$ . Let

$r(L_\infty)$ ,

$r'(L_\infty)$ and

$r''(L_\infty)$ denote the

$\lambda$ -invariants of the modules

$R(L_\infty)$ ,

$R'(L_\infty)$ and

$R''(L_\infty)$ , respectively.

In addition, we need one more invariant, denoted by

$d(L_\infty)$ , which is a

$\lambda$ -invariant of the module

$D(L_\infty)$ . This module is defined in the case where the field

$L_\infty$ is Abelian over

$\ell$ and

$k\subset L$ . The definition of

$D(L_\infty)$ can be found in the work [1], § 6, which also contains the definition of the modules

$V(L_\infty)$ ,

$V^+(L_\infty)$ and

$V^-(L_\infty)$ involved in the definition of

$D(L_\infty)$ , and a detailed description of the structure of this module and its properties can be found in [6]. Here we only note that

$D(L_\infty)$ is defined as

$V(L_\infty)/(V^+(L_\infty)\oplus V^-(L_\infty))$ , where

$V(L_\infty)$ ,

$V^+(L_\infty)$ and

$V^-(L_\infty)$ are free

$\Lambda$ -modules. It should also be noted that we often use without special mention the additive notation for the operation of multiplication since the operation of addition is not used here.

§ 3. An analogue of the Riemann–Hurwitz formula

This formula was first obtained by this author in [5]. A refined proof can be found in [6]. Let

$L'/L$ be a finite

$\ell$ -extension of algebraic number fields (here

$\ell$ is an arbitrary prime number). Let

$L'_\infty$ and

$L_\infty$ be cyclotomic

$\mathbb Z_\ell$ -extensions of the fields

$L'$ and

$L$ , respectively, where the fields

$L$ and

$L'$ satisfy some additional constraints, namely,

$L$ and

$L'$ are Abelian over

$\ell$ . This means that for any place

$v$ over

$\ell$ the completion

$L_v$ of the field

$L$ (or the completion

$L'_v$ of

$L'$ ) is an Abelian extension of the field

$\mathbb Q_\ell$ . We also assume that the field

$L$ contains a primitive

$\ell$ th root

$\zeta_0$ of unity, which means that

$L\supseteq k$ and all Galois modules considered below have zero Iwasawa

$\mu$ -invariants. The last condition holds for

$k_\infty$ when

$\ell=3$ , since 3 is a regular prime number, and therefore it holds for any finite

$\ell$ -extension of the field

$k_\infty$ . In particular, it holds for

$L_\infty$ and

$L'_\infty$ .

The analogue of the Riemann–Hurwitz formula provides a relation between the Iwasawa

$\lambda$ -invariants of the Galois modules defined in § 2 for the fields

$L_\infty'$ and

$L_\infty$ , where

$L'/L$ is a finite

$\ell$ -extension of algebraic number fields and the fields

$L'$ and

$L$ satisfy some additional conditions which were indicated above.

Remark. It is obvious that $g(k_\infty)=0$ . In the case $\ell=3$ an application of the above formula to the extension $K_\infty/k_\infty$ , where $K_\infty/k_\infty$ is a cyclic extension of degree $\ell$ such that there are precisely three places not over $\ell$ ramifying in $K_\infty/k_\infty$ , gives $g(K_\infty)=1$ .

As explained in [4], Proposition 3.1, a finite unramified

$\ell$ -extension

$L_\infty/K_\infty$ for

${\ell=3}$ obeys the relation

$g(L_\infty)=1$ and one of the following two options holds:

(A)

$d(L_\infty)=2(\ell-1)$ and

$\lambda'(L_\infty)=\lambda''(L_\infty)=r''(L_\infty)=0$ ;

(B)

$d(L_\infty)=0$ ,

$\lambda'(L_\infty)=\ell-1$ and

$\lambda''(L_\infty)=r''(L_\infty)=0$ .

$L_\infty/K_\infty$ is a finite unramified extension, then either both fields pertain to type (A), or both fields pertain to type (B). In this section we consider Case (A) in detail.

$L/K_n$ is a finite unramified Galois extension, then the Galois group

$G(L_\infty/K_n)$ is isomorphic to the direct product

$G(L/K_n)\times G(K_\infty/K_n)$ and the group

$G(L/K_n)$ acts on the

$\Lambda_n$ -modules

$D(L_\infty), T_\ell(L_\infty)$ and the other

$\Lambda$ -modules defined in § 2.

Proof. In Case (A) we have

$d(L_\infty)=d(K_\infty)=2(\ell-1)=4$ . The inclusion

$K_\infty\hookrightarrow L_\infty$ induces a natural mapping

$i\colon D(K_\infty)\to D(L_\infty)$ , and the sequence of the norm mappings

$N_m\colon L_m^\times\to K_m^\times$ , where

$L_m=K_m\cdot L$ , induces a mapping

$N\colon D(L_\infty)\to D(K_\infty)$ . It is obvious that

$N\circ i=[L:K_n]$ . Since

$D(K_\infty)$ and

$D(L_\infty)$ are free

$\mathbb Z_\ell$ -modules of the same rank, this means that

$i$ maps

$D(K_\infty)$ isomorphically onto a submodule of finite index in

$D(L_\infty)$ . The mappings

$i$ and

$N$ are

$G(L/K_n)$ -homomorphisms and the group

$G(L/K_n)$ acts identically on

$D(K_\infty)$ . Hence the group

$G(L/K_n)$ also acts identically on

$D(L_\infty)$ .

The proof of the proposition is complete.

Proof. It is sufficient to verify that the groups

$T_\ell(L_\infty)$ and

$R_\ell(L_\infty)$ are finite. The finiteness of the group

$T_\ell(L_\infty)$ follows from the fact that in Case (A) we have

$\lambda'(L_\infty)=\lambda''(L_\infty)=0$ .

Since in Case (A) we have $r''(L_\infty)=0$ , to prove that the group $R_\ell(L_\infty)$ is finite it is sufficient to establish the equality $r'(L_\infty)=0$ . Note that all the places of the field $L$ over $\ell$ are purely ramified in the extension $L_\infty/L$ , and therefore the group $\Gamma_n=G(L_\infty/L)$ acts identically on $R_\ell(L_\infty)$ and, by implication, on $R'(L_\infty)$ . Since there exists an epimorphism $\operatorname{Tors}X(L_\infty)\to R'(L_\infty)$ , it is sufficient to verify that

$\begin{equation*} \operatorname{rk}\bigl(\operatorname{Tors}X(L_\infty)/(\gamma_n-1) \operatorname{Tors}X(L_\infty)\bigr)=0, \end{equation*} \notag$

where $\gamma_n=\gamma_0^{\ell^n}$ is a topological generator of the group $\Gamma_n$ . If this rank were positive, then the field $L$ would have ‘superfluous’ $\Gamma$ -extensions, which would mean that it disobeys the Leopoldt conjecture, contrary to the finiteness of the module $T_\ell(L_\infty)$ (see [1], Proposition 5.1, or [7], Theorem 4.1).

The proof of the proposition is complete.

$L$ is an extension of the field

$K_n$ such that

$L\cap K_\infty=K_n$ , then all Galois modules connected with the extension

$L_\infty/L$ are

$\Gamma_n$ -modules or

$\Lambda_n$ -modules, where

$\Lambda_n=\mathbb Z_\ell[[\Gamma_n]]$ .

Let

$\mathscr F(V(L_\infty)/V^+(L_\infty))$ be a minimal free

$\Lambda_n$ -module containing the torsion-free

$\Lambda_n$ -module

$V(L_\infty)/V^+(L_\infty)$ as a submodule of finite index. Then there exists a natural embedding of the free

$\Lambda_n$ -module

$V^-(L_\infty)$ into the free module

$\mathscr F(V(L_\infty)/V^+(L_\infty))$ and the quotient module

$D'(L_\infty)=\mathscr F(V(L_\infty)/V^+(L_\infty))/V^-(L_\infty)$ has no nonzero finite submodules, which means that it is a free

$\mathbb Z_\ell$ -module containing

$D(L_\infty)$ as a submodule of finite index. Consequently, the Galois group

$G(L/K_n)$ also acts identically on

$D'(L_\infty)$ . Set

$E'(L_\infty)=D'(L_\infty)/D(L_\infty)$ . Then the group

$G(L/K_n)$ acts identically on

$E'(L_\infty)$ .

In a similar way we set

$D''(L_\infty)=\mathscr F(V(L_\infty)/V^-(L_\infty))$ and

$E''(L_\infty)=D''(L_\infty)/D(L_\infty)$ . Again, we show that the group

$G(L/K_n)$ acts identically on

$E''(L_\infty)$ . The groups

$E'(L_\infty)$ and

$E''(L_\infty)$ are isomorphic as Abelian groups (the isomorphism is induced by the skew automorphism

$\psi$ ; [1], Theorem 6.1). In addition,

$D'(L_\infty)\cap D''(L_\infty)= D(L_\infty)$ ([6], Proposition 1.7), and therefore the group

$E'(L_\infty)\oplus E''(L_\infty)$ is embedded into

$(D(L_\infty)\otimes \mathbb Q_\ell)/D(L_\infty)$ and the minimum number of generators of the group

$E'(L_\infty)$ does not exceed

$\ell-1=2$ .

Proof. By Theorem 3.1 in [2] the Galois module

$E'(L_\infty)$ contains a submodule

$E'_2(L_\infty)$ naturally isomorphic to the module

$\overline T_\ell(L_\infty)$ (note that the proof of that theorem is based only of the fact that

$\overline T_\ell(L_\infty)$ is finite, so its claim is not only true for

$K_\infty$ , but also for the field

$L_\infty$ ). Then it follows from Proposition 2 that

$G(L/K_n)$ acts identically on

$\overline T_\ell(L_\infty)$ .

Since the extension $L_\infty/L$ is purely ramified in the divisors of $\ell$ , the natural mapping $\overline T_\ell(L_\infty)\to \operatorname{Cl}_\ell(L)$ is epimorphic. Consequently, $G(L/K_n)$ also acts identically on $\operatorname{Cl}_\ell(L)$ .

The proof of is complete.

§ 4. The Riemann–Hurwitz formula (Case (B))

Proof. Let

$H :=G(K_\infty/k_\infty)$ . Then

$T_\ell(K_\infty)$ is a cyclic

$H$ -module ([1], Theorem 4.1) annihilated by the operator

$N_H$ . With regard to the condition

$\lambda'(L_\infty)=2$ this means that

$T_\ell(K_\infty)\cong \mathbb Z_\ell^2$ as a

$\mathbb Z_\ell$ -module. Suppose that

$\gamma_0$ acts on roots of unity

$\zeta_n$ by the rule

$\gamma_0(\zeta_n)=\zeta_n^{\varkappa(\gamma_0)}$ ,

$\varkappa(\gamma_0)\in\mathbb Z_\ell$ . Then by [1], Theorem 5.1,

$\gamma_0$ acts on

$T_\ell(K_\infty)$ by multiplication by

$\sqrt{\varkappa(\gamma_0)}$ . In the case

$\ell=3$ we assume that

$\gamma_0$ is defined by the condition

$\varkappa(\gamma_0)=4$ , where

$\sqrt{\varkappa(\gamma_0)}=-2$ .

Then, as in the proof of Proposition 1, the embedding of fields $K_\infty\hookrightarrow L_\infty$ induces the embedding $i\colon T_\ell(K_\infty)\hookrightarrow T_\ell(L_\infty)$ with a finite cokernel. Let $T_\ell^0(L_\infty)$ be the maximal $\mathbb Z_\ell$ -free quotient module of the module $T_\ell(L_\infty)$ . Then $T_\ell^0(L_\infty)$ is acted upon by the topological generator $\gamma_n=\gamma_0^{\ell^n}$ of the group $\Gamma_n$ and $\gamma_n$ multiplies $T_\ell^0(L_\infty)$ by $-2^{\ell^n}$ . The group $G(L/K_n)$ acts identically on $T_\ell^0(L_\infty)$ .

As in the proof of Proposition 2, we use Theorem 4.1 in [7] to show that any field $L_n$ obeys the Leopoldt conjecture and thus $r'(L_\infty)=0$ and $R(L_\infty)$ is a finite group. However, by [4], Proposition 3.3, the module $\overline T_\ell(L_\infty)$ has no nontrivial finite submodules. This means that $\overline T_\ell(L_\infty)=T_\ell(L_\infty)=T_\ell^0(L_\infty)\cong \mathbb Z_3^2$ .

The proof of the proposition is complete.

Proof. There is a natural mapping

$\varphi\colon \overline T_\ell(L_\infty)\!\to\! \operatorname{Cl}_\ell(L)$ , which is a

$G(L/K_n)$ -homomorphism. Since the extension

$L_\infty/L$ is purely ramified at the places lying over

$\ell$ ,

$\varphi$ is epimorphic. By Proposition 4 the group

$G(L/K_n)$ acts identically on

$\overline T_\ell(L_\infty)$ . Consequently, it also acts identically on

$\operatorname{Cl}_\ell(L)$ as well.

The proof of the proposition is complete.

To use Propositions 3 and 5 in the proof of our theorem we need the following simple statement concerning finite

$\ell$ -groups.

Proof. Assume that

$G$ is not Abelian, that is,

$[G,G]\!\neq\! 1$ . Then

${[G,G]/[G,[G,G]]\!\neq\! 1}$ . Let

$\varphi$ be an epimorphism of the group

$[G,G]/[G,[G,G]]$ onto

$A\cong \mathbb F_\ell$ . Then the group extension

$\begin{equation*} 1\to [G,G]\to G\to G^{\mathrm{ab}}\to 1, \end{equation*} \notag$

where

$G^{\mathrm{ab}}=G/[G,G]$ , and the epimorphism

$\varphi$ induce the central extension

$\begin{equation*} 1\to A\to B\to G^{\mathrm{ab}}\to 1, \end{equation*} \notag$

where

$B=G/\ker \varphi$ . Thus,

$[B,B]=A$ and there exist elements

$x,y\in B$ such that

$[x,y]\neq 1$ . Let

$B_1$ be the subgroup of

$B$ generated by

$x^\ell$ and

$y$ . Then

$B_1$ is an Abelian subgroup and

$x$ acts nontrivially on

$B_1$ . Set

$F=B/B_1$ . Then there exists a group extension

$\begin{equation*} 1\to B_1\to B\to F\to 1 \end{equation*} \notag$

such that

$F$ acts nontrivially on the Abelian kernel

$B_1$ .

The group $F$ is a quotient group of $G$ with some kernel $H$ , hence there exists a commutative diagram of group extensions

Consequently,

$F$ acts nontrivially on

$H/[H,H]$ , contrary to the hypothesis of the proposition.

The proof of the proposition is complete.

Proof of the theorem. Assume that there exists a finite unramified Galois

$\ell$ -extension

$M/K_n$ with non-Abelian Galois group

$G$ . Suppose that the subfield

$L\subset M$ is a Galois extension, that is, the group

$H=G(M/L)$ is a normal subgroup of

$G$ . Set

$L_1=M^{[H,H]}$ . Then

$L_1$ is an Abelian unramified

$\ell$ -extension of the field

$L$ and, consequently, there exists a canonical epimorphism

$\operatorname{Cl}_\ell(L)\to G(L_1/L)=H/[H,H]$ .

The Galois group $G(L/K_n)$ acts identically on the group $\operatorname{Cl}_\ell(L)$ by Propositions 3 and 5, and therefore it also acts identically on $H/[H,H]$ . Thus, all the hypotheses of Proposition 6 are satisfied, and so the group $G$ is Abelian, which means that there can exist only Abelian unramified $3$ -extensions over $K_n$ . In turn, this means that the claim of the theorem holds for $K_n$ .

§ 5. The form of the field

$K$ and the structure of the class groups of the fields

$K_n$

For the reader’s convenience we give a brief survey of the results obtained in [1]–[4] and concerning the field

$K$ and the class groups

$\operatorname{Cl}_\ell(K_n)$ .

$\ell=3$ and

$K$ is a cubic extension of the field

$k$ such that

$K$ is a Galois extension of

$\mathbb Q$ with Galois group

$S_3$ ,

$K$ is Abelian over

$\ell$ , and there are precisely three places not over

$\ell$ that ramify in the extension

$K_\infty/k_\infty$ , then

$K$ pertains to one of the following three types, which, as in [1], are referred to as Cases 2.1, 2.2, and 2.4 (see [1], Propositions 2.1, 2.2 and 2.4). The work [1] also contains one more case, namely, Case 2.3, but it cannot occur for

$\ell=3$ .

Case 2.1.

$K=k(\sqrt[3]{a})$ ,

$a\in\mathbb Z$ ,

$a^{\ell-1}\equiv 1\pmod{\ell^2}$ and

$a=p_1^{r_1}p_2^{r_2}p_3^{r_3}$ , where

$p_1$ ,

$p_2$ and

$p_3$ are distinct prime numbers not equal to

$\ell$ . Here

$r_1r_2r_3\not\equiv 0\pmod\ell$ and the principal divisors

$(p_i)$ remain prime in

$K_\infty$ for

$i=1,2,3$ . The last condition means that the

$p_i$ are primitive roots modulo

$\ell^2$ .

In Case 2.1 the module

$\overline T_\ell(K_\infty)$ starts with

$\mathbb F_\ell(1)$ , and there can be three subcases.

Subcase 2.1.a.

$|T_\ell(K_\infty)|=3, |\overline T_\ell(K_\infty)|=27$ ,

$\operatorname{Cl}_\ell(K)\cong (\mathbb Z/3\mathbb Z)^2$ and

$\operatorname{Cl}_\ell(K_n)\cong \mathbb Z/3\mathbb Z\oplus \mathbb Z/9\mathbb Z$ for

$n>0$ .

Subcase 2.1.b.

$|T_\ell(K_\infty)|=3^r$ ,

$|\overline T(K_\infty)|=3^{r+2}$ for some odd

$r>1$ , and

$\operatorname{Cl}_\ell(K_n)\cong (\mathbb Z/3^{n+1})^2$ for

$n<n_0=(r-1)/2$ and

$\operatorname{Cl}_\ell(K_n)\cong \mathbb Z/3^{n_0+1}\mathbb Z\oplus \mathbb Z/3^{n_0+2}\mathbb Z$ for

$n\geqslant n_0$ .

Subcase 2.1.c.

$T_\ell(K_\infty)\cong \mathbb Z_3^2$ and

$\operatorname{Cl}_\ell (K_n)\cong (\mathbb Z/3^{n+1}\mathbb Z)^2$ for

$n\geqslant 0$ .

For any pair of distinct prime numbers

$p_1$ ,

$p_2$ there exist infinitely many primes

$p_3$ such that

$K$ pertains to Subcase 2.1.a, and infinitely many primes

$p_3$ such that

$K$ pertains either to Subcase 2.1.b, or to Subcase 2.1.c, though there is no field

$K$ of the last type that can definitely be classified as pertaining to Subcase 2.1.b or to Subcase 2.1.c (see [2], Theorem 4.1 and Proposition 4.3, and [3], Theorem 6.2).

Case 2.2. In this case

$K=k(\sqrt[3]{a})$ ,

$a^{\ell-1}\equiv 1\pmod{\ell^2}$ and

$a=p^r q^s$ ,

$rs\not\equiv 0\ (\operatorname{mod}\ell)$ , where

$p$ and

$q$ are distinct prime numbers not equal to

$\ell$ , and

$p$ splits into a product of two divisors

$\mathfrak p_1$ and

$\mathfrak p_2$ in the unique quadratic subfield

$F$ of

$k$ (note that for

$\ell=3$ the fields

$F$ and

$k$ coincide) such that each divisor remains prime in the extension

$K_\infty/F$ .

The divisor

$(q)$ remains prime in the extension

$K_\infty/\mathbb Q$ . In this case

$\overline T_\ell(K_\infty)$ starts with

$\mathbb F_\ell(0)$ and the following three subcases are possible.

Subcase 2.2.a.

$T(K_\infty)=0$ and

$\overline T_\ell(K_\infty)\cong (\mathbb Z/3\mathbb Z)^2$ . Then

$\operatorname{Cl}_\ell(K_n)\cong (\mathbb Z/3\mathbb Z)^2$ for every

$n$ .

Subcase 2.2.b.

$T_\ell(K_\infty)\cong (\mathbb Z/3^{n_0}\mathbb Z)^2$ for some index

$n_0$ and

$|R_\ell(K_\infty)|=9$ . In this subcase

$\operatorname{Cl}_\ell(K_n)\cong (\mathbb Z/\ell^{n+1}\mathbb Z)^2$ for

$n\leqslant n_0$ and

$\operatorname{Cl}_\ell(K_n)\cong (\mathbb Z/\ell^{n_0}\mathbb Z)^2$ for

${n>n_0}$ .

Subcase 2.2.c.

$T_\ell(K_\infty)\cong \mathbb Z_3^2$ . In this case we have

$\operatorname{Cl}_\ell(K_n)\cong (\mathbb Z/3^{n+1}\mathbb Z)^2$ for every

$n$ .

Again, it is known that there exist infinitely many fields

$K$ pertaining to Subcase 2.2.a and infinitely many fields

$K$ pertaining to Subcases 2.2.b and 2.2.c, but there is no field of the last type that can definitely be classified as pertaining to Subcase 2.2.b or to Subcase 2.2.c (see [4], Propositions 4.1–4.4). Note also that in Case 2.2 there exist fields

$K$ pertaining to none of the three subcases distinguished above.

Case 2.4. In this case we have

$K=k(\sqrt[3]{p})$ , where the prime number

$p$ satisfies the congruence

$p\equiv 8,17\pmod{27}$ . This case remains unexplored so far, and there is nothing definite about it yet.

Citation in format AMSBIB

\Bibitem{Kuz24}

\by L.~V.~Kuz'min

\paper On a~family of algebraic number fields with finite 3-class field tower

\jour Sb. Math.

\yr 2024

\vol 215

\issue 7

\pages 911--919

\mathnet{http://mi.mathnet.ru/eng/sm10040}

\crossref{https://doi.org/10.4213/sm10040e}

\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4813933}

\zmath{https://zbmath.org/?q=an:07945701}

\adsnasa{https://adsabs.harvard.edu/cgi-bin/bib_query?2024SbMat.215..911K}

\isi{https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=Publons&SrcAuth=Publons_CEL&DestLinkType=FullRecord&DestApp=WOS_CPL&KeyUT=001346292600003}

\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85208137328}

§ 1. Introduction

§ 2. Notation and definitions

§ 3. An analogue of the Riemann–Hurwitz formula

§ 4. The Riemann–Hurwitz formula (Case (B))

§ 5. The form of the field KK and the structure of the class groups of the fields K_nK_n

Bibliography

§ 5. The form of the field $K$ and the structure of the class groups of the fields $K_n$