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On a family of algebraic number fields with finite 3-class field tower L. V. Kuz'min National Research Centre "Kurchatov Institute", Moscow, Russia
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     § 1. IntroductionLet $L$ 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 $K$. The main result of this work consists in the proof of the following theorem. 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 definitionsWe 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 formulaThis 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. For $L_\infty$ we set
$$
\begin{equation*}
g(L_\infty)=\frac{1}{2} d(L_\infty)+\lambda'(L_\infty)+2\lambda''(L_\infty)+2r''(L_\infty).
\end{equation*}
\notag
$$
The analogue of the Riemann–Hurwitz formula relates the invariant $2g(L_\infty)$ to an invariant $2g(L'_\infty)$ defined similarly for the finite $\ell$-extension $L'/L$. Namely, under the above assumptions about the fields $L_\infty$ and $L'_\infty$ the following relation, which is an analogue of the Riemann–Hurwitz formula, holds:
$$
\begin{equation*}
2g(L'_\infty)-2=[L'_\infty:L_\infty][2g(L_\infty)-2]+\sum_{v\nmid\ell}(e_v-1),
\end{equation*}
\notag
$$
where $v$ ranges over all the places of the field $L_\infty'$ that are not over $\ell$ and $e_v$ means the ramification index of the place $v$ in the extension $L'_\infty/L_\infty$. 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 $\lambda''(L_\infty)=r''(L_\infty)=0$. If $L_\infty/K_\infty$ 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. If $L/K_n$ 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. If $L$ 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))Assume that the field $L$ satisfies the conditions of Case (B). Proposition 4. In Case (B) $\overline T_\ell(L_\infty)=T_\ell(L_\infty)$ and $T_\ell(L_\infty)\cong\mathbb Z_3^2$. 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. Proposition 5. In Case (B) the group $G(L/K_n)$ acts identically on the group $\operatorname{Cl}_\ell(L)$. 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. Proposition 6. Let $G$ be a finite $\ell$-group such that for each normal subgroup $H$ of $G$ the quotient group $G/H$ acts identically on the $G/H$-module $H^{\mathrm{ab}}=H/[H,H]$. Then $G$ is an Abelian group. 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 where $G^{\mathrm{ab}}=G/[G,G]$, and the epimorphism $\varphi$ induce the central extension 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 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 $\operatorname{Cl}_\ell(K_n)$. If $\ell=3$ 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. 
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