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This article is cited in 2 scientific papers (total in 2 papers) Arithmetic of certain L. V. Kuz'min National Research Centre "Kurchatov Institute", Moscow
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 6, DOI: https://doi.org/10.4213/im9241 
Bibliographic databases:
    § 1. IntroductionLet $\ell$ be a regular odd prime number, $k=\mathbb Q(\zeta_0)$, where $\zeta_0$ is a primitive root of unity of degree $\ell$ and $K=k(\sqrt[\ell]{a}\,)$, where $a$ is a natural number of the form (3.1) such that its prime divisors $p_1$, $p_2$, $p_3$ remain prime in the cyclotomic $\mathbb Z_\ell$-extension $k_\infty$ of $K$. The arithmetic of $K$, which has many interesting features, was a subject of our investigation in [1] and [2]. So, it appeared that the $\ell$-component of the class group $\operatorname{Cl}(K)_\ell$ of $K$ is non-zero. It has an order at most $\ell^{\ell-1}$, and its period is $\ell$. In the simplest case $\ell=3$ there were found in empirical way the fields $K$, such that their places over $\ell$, generate a subgroup of order $\ell$ in the class group $\operatorname{Cl}(K)_\ell$ (the fields of the type A1 in terminology of [2]) and the fields such that all their places over $\ell$ are principal divisors (the fields of the type A2). Essentially nothing was known in the case $\ell>3$, though it was proved in [1] that in the case of infinite Tate module $T_\ell(K_\infty)$ of $K_\infty$ the generator $\gamma_0$ of the Galois group $\Gamma=G(K_\infty/K)$, which acts on the roots of unity of the $\ell$-primary degree by the rule $\gamma_0(\zeta_n)=\zeta_n^{\varkappa(\gamma_0)}$, where $\varkappa(\gamma_0)\in \mathbb Z_\ell$, acts on $T_\ell(K_\infty)$ by multiplication by $\sqrt{\varkappa(\gamma_0)}$ [1], Theorem 5.1. It was proved in [2] that in the case of finite Tate module $T_\ell(K_\infty)$ there is some greater group $E'(K_\infty)$, which contains $T_\ell(K_\infty)$, such that $\gamma_0$ acts on it also by multiplication by $\sqrt{\varkappa(\gamma_0)}$. In the present paper we use a new method to study the arithmetic of $K$. Namely, simultaneously with $K$ we consider another field $L$ of the form (3.2) such that there are only two places $p_1$ and $p_2$ ramified in $L/k$. The arithmetic of $L$ is simpler than that of $K$. Moreover, we can fix the field $L$ and change a prime number $p_3$. Thus, we consider a family of the fields $K=K(p_3)$, and this consideration leads to a series of interesting consequences. In § 3 we consider the Galois cohomologies groups $H^i(G(L/k),U(L))$ and $H^i(G(K/k), U(K))$. We define some critically important subgroup $\overline U_2(L)$ or $\overline U_2(K)$ of the group of units $U(L)$ or $U(K)$ and prove that $\overline U_2(L)$ is a cohomologically trivial module. Concerning the field $K$, in the present moment we cannot say whether the module $\overline U_2(K)$ may be not cohomologically trivial. In the present paper we use only results that take place for $L$, but since the proofs for $L$ and for $K$ are just the same, we give both proofs, thou the results for $K$ will be used only in the next our paper. In § 4 we consider some general properties of $G$-modules, where $G=G(K/\mathbb Q)$ and $G=\widetilde G=G(L/\mathbb Q)$. We apply these results for characterization of the Galois module $\mathcal{A}(L)/\mathcal{A}(L)^\ell$ (Theorem 4.1), where $\mathcal{A}(L)=\prod_{v|\ell}U^{(1)}(L_v)$ and $U^{(1)}(L_v)$ is the group of principal units of the local field $L_v$. In § 5 we apply the obtained results for characterization of some Abelian extensions of the field $L$. Namely, let $N=N(p_3)$ be the maximal Abelian $\ell$-extension of $L$ such that $N\supset K$, $N/K$ is unramified, and the Galois group $G(N/L)$ is of period $\ell$. Using the obtained information about the group $\mathcal{A}(L)/\mathcal{A}(L)^\ell$, we can calculate the Kummer group of the extension $N/L$, and thus, determine its degree. It appears that always we have $[N:L]=\ell^r$ with an even $r$ and $2\leqslant r\leqslant \ell-1$. At last, using the Chebotarev density theorem, we prove that any possible value of $r$ realize for infinitely many values of $p_3$. If $r<\ell-1$ then all the places over $\ell$ split completely in the extension $N/L$. In the case $r=\ell-1$, using the Chebotarev theorem, we prove that there are infinitely many $p_3$ such that $r=\ell-1$ and all places over $\ell$ split completely in $N/L$ and infinitely many $p_3$ such that $r=\ell-1$ and all places over $\ell$ have inertia degree $\ell$ in $N/L$. In § 6 we apply the results obtained for the extensions $N/L$ to the extensions of $K$. We prove that for any prime $p_3$ the order of the $\ell$-class group of $K=K(p_3)$ is at least $\ell^2$ (Theorem 6.1). In the case $\ell=3$ we prove that there are infinitely many $p_3$ such that $K(p_3)$ is of the type A1 and infinitely many $p_3$ such that $K(p_3)$ is of the type A2. § 2. Notations and definitionsWe try to follow the notations of [1], [2]. For a regular odd prime $\ell$, let $\zeta_n$ be a primitive root of unity of degree $\ell^{n+1}$. Let $k=\mathbb Q(\zeta_0)$, $k_n=\mathbb Q(\zeta_n)$ and $k_\infty=\bigcup_{n=1}^\infty k_n$ be the cyclotomic $\mathbb Z_\ell$-extension of $k$. The prime number $p\neq\ell$ remains prime in the field $k_\infty$ if and only if $p$ is a primitive root modulo $\ell^2$. We denote the set of all such primes $p$ by $\mathbb P_0$. We consider two fields $K$ and $L$ defined by (3.1) and (3.2), respectively. We put $H=G(K/k)$ and $\widetilde H=G(L/k)$, where $G(K/k)$ means the Galois group of $K/k$. Thus, $K$ and $L$ are Galois extensions of degree $\ell(\ell-1)$ with isomorphic Galois groups $G=G(K/\mathbb Q)$ and $\widetilde G=G(L/\mathbb Q)$. The Galois group $\Delta=G(k/\mathbb Q)$ acts on $H$ (or on $\widetilde H$) via the Teichmüller character $\omega\colon\Delta\to(\mathbb Z/\ell\mathbb Z)^\times$. By groups of cohomologies we mean everywhere the Tate groups if the opposite is not said explicitly. We denote by $S$ the set of all places over $\ell$ of the field under consideration. By $\operatorname{Cl}(K)_\ell$ we denote the $\ell$-component of the class group of $K$, and $\operatorname{Cl}_S(K)_\ell$ means a factorgroup of $\operatorname{Cl}(K)_\ell$ by the subgroup generated by all places in $S$. Let $\overline{\mathbf N}$ be the maximal Abelian unramified $\ell$-extension of $K_\infty$ and $\mathbf{N}$ the maximal subfield of $\overline{\mathbf N}$ such that all places over $\ell$ split in $\mathbf{N}/K_\infty$. We denote the Galois groups of $\overline{\mathbf N}/K_\infty$ and $\mathbf{N}/K_\infty$ by $\overline T_\ell(K_\infty)$ and $T_\ell(K_\infty)$ respectively. These groups are modules with respect to the action of the Galois group $\Gamma=G(K_\infty/K)$. For a field $K$ or its completion $K_v$, we denote by $U(K)$ and $U(K_v)$ the unit group of the corresponding field. By $\mu_\ell(K)$ we denote the group of all $\ell$-primary roots of unity in $K$. For a local field $K_v$, we denote by $U^{(1)}(K_v)$ the group of principal units in $K_v$. By $D(K)$ we denote the group of divisors of $K$. For an Abelian group $A$, we denote by $A[\ell]$ the pro-$\ell$-completion of $A$. By $\overline U(K)$ we denote the group $U(K)/\mu(K)$, where $\mu(K)$ is the group of all roots of unity in $K$. By $\mathcal{A}(K)$ we denote the group $\prod_{v\in S} U^{(1)}(K_v)$ and by $\overline{\mathcal{A}}(K)$ the group $\prod_{v\in S}\overline U^{\,(1)}(K_v)$, where $\overline U^{\,(1)}(K_v)=U^{(1)}(K_v)/\mu_\ell(K_v)$ and $\mu_\ell(K_v)$ is the $\ell$-component of $\mu(K_v)$. We shall often use additive notations for multiplication without special reminding, since we do not use addition in this paper. § 3. Cohomologies of certain groups of unitsSo, let $p_1,p_2,p_3\in \mathbb P_0$ and $K=k(\sqrt[\ell]{a}\,)$, where
$$
\begin{equation}
a=p_1^{r_1}p_2^{r_2}p_3^{r_3},\qquad r_1r_2r_3\not\equiv 0\pmod \ell,\qquad a^{\ell-1}\equiv 1\pmod{\ell^2}.
\end{equation}
\tag{3.1}
$$
Then $K/k$ is an extension described in [1], Proposition 2.1. This extension ramifies at $\mathfrak {p}_1$, $\mathfrak {p}_2$ and $\mathfrak {p}_3$, where $\mathfrak{p}_i=(p_i)$ for $i=1,2,3$, and any place $v$ of $K$ over $\ell$ splits in this extension. The analogous result holds also for any $n$ for the extension $K_n/k_n$, where $K_n=K\cdot k_n$. The places $\mathfrak p_i$ remain prime in $k_n$ and ramify in the extension $K_n/k_n$. In $k_n$ there is the single place $v_n$ over $\ell$, and $v_n$ splits completely in $K_n/k_n$. In parallel with $K$ we shall consider the field $L\colon =k(\sqrt[\ell]{b}\,)$, where
$$
\begin{equation}
b=p_1^{s_1}p_2^{s_2},\qquad s_1s_2\not\equiv 0\pmod{\ell}, \qquad b^{\ell-1}\equiv 1\pmod{\ell^2}.
\end{equation}
\tag{3.2}
$$
The extension $L/k$ ramify in $\mathfrak p_1$, $\mathfrak p_2$, and the place $v$ splits completely in it. As it was shown in [1], Proposition 2.1, for a given triple of primes $p_1,p_2,p_3\in \mathbb P_0$ there are $\ell-2$ different fields $K$ corresponding to this choice of $r_1$, $r_2$, $r_3$. On the contrary, the numbers $p_1$, $p_2$ define $L$ uniquely. The next assertion is an analogue of Theorem 4.1 of [1]. Proposition 3.1. The following equalities hold: $\overline T_\ell(L_\infty)=T_\ell(L_\infty)=0$, $\operatorname{Cl}(L)_\ell=\operatorname{Cl}_S(L)_\ell=0$. Proof. Consider an $\widetilde H$-module $\overline T_\ell (L_\infty)$. Let $f$ be the minimal number of its generators as $\widetilde H$-module. Then $f$ equals the minimal number of generators of the Abelian group $C:=\overline T_\ell(L_\infty)/(\widetilde\sigma-1)\overline T_\ell(L_\infty)$. As in [1], this implies that $C$ is isomorphic to $\mathbb F^f_\ell$ as an Abelian group.
Let $\overline N$ be the extension of $L_\infty$, whose Galois group is naturally isomorphic to the group $\overline T_\ell(L_\infty)$. Let $\overline N_0$ be the subfield of $\overline N$ fixed by the action of the Galois group $(\widetilde\sigma-1)\overline T_\ell(L_\infty)$, where $\widetilde \sigma$ is a generator of the group $\widetilde H$. Then, as in [1], we obtain that $G(\overline N_0/k_\infty)$ is an Abelian group of period $\ell$ and of order $\ell^{f+1}$. To calculate $f$, we use the Kummer theory. Let $M$ be a subfield of $\overline N_0$ of degree $\ell$ over $L_\infty$. Then $M$ is of the form $M=k_\infty(\sqrt[\ell]{\alpha}\,)$. Applying to the field $M$ the same arguments that were used in the proof of Theorem 4.1 of [1], we get that the element $\alpha$ is defined uniquely up to raising in some power prime to $\ell$ and multiplication by some $\ell$-th power. It means that $M=\overline N_0$. Then $\overline N_0=L_\infty$ and $f=0$. Therefore, $\overline T_\ell(L_\infty)=0$. Since there are epimorphisms $\overline T_\ell(L_\infty)\to T_\ell(L_\infty)$, $\overline T_\ell(L_\infty)\to\operatorname{Cl}(L)_\ell$ and $\overline T_\ell(L_\infty)\to\operatorname{Cl}_S(L)_\ell$, we get $T_\ell(L_\infty)=0$ and $\operatorname{Cl}(L)_\ell=\operatorname{Cl}_S(L)_\ell=0$. This proves Proposition 3.1. Remark. We can give more short but less direct proof of Proposition 3.1. Namely, we shall prove that $\overline T_\ell(L_\infty)=0$. Applying the Riemann–Hurwitz formula to the extension $L_\infty/k_\infty$ (see, for example, [1], Theorem 1.1), we obtain $d(L_\infty)=\lambda'(L_\infty)=\lambda''(L_\infty)=r''(L_\infty)=0$. Therefore, the module $\overline T_\ell(L_\infty)$ is at most finite and $D(L_\infty)=0$ (the definition of $D(L_\infty)$ one can find in [1], § 6). Then the module $E'(L_\infty)$ defined in [1], Proposition 6.3, also vanishes. According to [2], Theorem 3.1, the module $E'(L_\infty)$ contains a submodule $E'_2(L_\infty)$, which is isomorphic to $\overline T_\ell(L_\infty)$. Hence, $\overline T_\ell(L_\infty)=0$. Our nearest goal is to study the groups of cohomologies $H^i(H,U(K))$ and $H^i(\widetilde H,U(L))$, and also the cohomologies of some subgroups and factorgroups of these groups of units. The exact sequence of $H$-modules where $D_K^0$ is the group of principal divisors of $K$, induces an exact sequence of cohomologies
$$
\begin{equation}
1\to U(k)\to k^\times\to (D_K^0)^H\xrightarrow{\eta} H^1(H,U(K))\to 1.
\end{equation}
\tag{3.3}
$$
By Theorem 90 of Hilbert any element of $H^1(H,U(K))$ may be presented by a cocycle $f\colon H\to U(K)$ of the form $f=f_c$, where $f_c(h)=h(c)/c$, $c\in K^\times$ and the principal divisor $(c)$ is fixed under the action of $H$. Any such $(c)$ is of the form $(c)=a_1a_2$, where $a_1\in D(k)$ and $a_2$ is a product of some prime divisors of $K$ ramified in $K/k$. In particular, the group $H^1(H,U(K))$ contains an element, which is presented by the cocycle $f_{\overline a\,}(h)$, where $\overline a=\sqrt[\ell]{a}$ and $a$ is as in (3.1). Analogously, the group $H^1(\widetilde H, U(L))$ contains an element presented by $f_{\overline b\,}(h)$, where $\overline b=\sqrt[\ell]{b}$ and $b$ is the element in (3.2). These cocycles take their values in the groups $\mu_\ell(K)$ and $\mu_\ell(L)$ respectively. Proposition 3.2. The inclusion $\mu_\ell(K)\hookrightarrow U(K)$ induces inclusions Analogously, the inclusion $\mu_\ell(L)\hookrightarrow U(L)$ induces inclusions Proof. The class $\operatorname{cls}(f_{\overline a\,}(h))$ of the cocycle $f_{\overline a\,}(h)$ generates the group $H^1(H,\mu_\ell(K))$. Suppose that $i_1(\operatorname{cls}(f_{\overline a\,}(h)))=0$. It means that there is a unit $u\in U(K)$ such that $h(u)/u=h(\overline a)/\overline a$ for any $h\in H$. Then $u/\overline a\in k^\times$, $u_1:=u^\ell\in k$ and $K=k(\sqrt[\ell]{u}\,)$, but this is impossible. Indeed, the place $v$ of $K$ over $\ell$ splits completely in $K/k$, but any unit $u_1$ of $K$, which is an $\ell$-th power in the local field $k_v$, is also an $\ell$-th power in $K$ (otherwise $k(\sqrt[\ell]{u_1}\,)/k$ would be an unramified extension of degree $\ell$). This proves that the map $i_1$ in (3.4) is an inclusion. The monomorphic character of $j_1$ in (3.5) may be proved in the same way.
To prove the monomorphic character of $i_2$ in (3.4), it is enough to check that $\zeta_0$ is not a norm in the extension $K/k$. To do this, we note that $\mathfrak p_1=(p_1)$ ramifies in $K/k$ and also in the extension of local fields $K_{\mathfrak P_1}/k_{\mathfrak p_1}$, where $\mathfrak P_1$ is a prime divisor of $\mathfrak p_1$ in $K$. By local class field theory it means that
$$
\begin{equation}
\bigl(U(k_{\mathfrak p_1}):N(U(K_{\mathfrak P_1}))\bigr)=\ell,
\end{equation}
\tag{3.6}
$$
where $N$ means the norm map from $K_{\mathfrak P_1}$ into $k_{\mathfrak p_1}$. Since $\mathfrak p_1$ remains prime in $k_\infty$, the field $k_{\mathfrak p_1}$ does not contain the primitive root of unity $\zeta_1$ of degree $\ell^2$, that is, the $\ell$-component of $U(k_{\mathfrak p_1})$ is generated by $\zeta_0$. Then (3.6) means that $\zeta_0$ is not a norm in $K_{\mathfrak P_1}/k_{\mathfrak p_1}$ and, moreover, in $K/k$. This proves the monomorphic character of $j_1$ in (3.5). The monomorphic character of $j_2$ may be proved in the same way. This proves Proposition 3.2. For local or global algebraic number field $F$ we shall denote by $\overline U(F)$ the factor group $U(F)/\mu(F)$. Proof. The exact sequence of $H$-modules induces an exact sequence of cohomologies
$$
\begin{equation}
\begin{aligned} \, &\cdots\to H^0(H,\mu(K))\xrightarrow{\alpha}H^0(H,U(K))\to H^0(H,\overline U(K))\to H^1(H,\mu(K)) \nonumber \\ &\qquad\xrightarrow{\beta}H^1(H,U(K))\to H^1(H,\overline U(K))\xrightarrow{\gamma} H^2(H,\mu(K))\to\cdots\,. \end{aligned}
\end{equation}
\tag{3.9}
$$
Since $H^i(H,\mu(K))=H^i(H,\mu_\ell(K))$ for any $i$ it follows from Proposition 3.2 that $\alpha$ and $\beta$ are injections. So, for any $i$ and any $H$-module $A$ there is a natural isomorphism $H^i(H,A)\cong H^{i+2}(H,A)$. Hence the sequence (3.9) is periodic with period $2$, and injectivity of $\alpha$ yields $\gamma=0$. Therefore, (3.9) induces short exact sequences
$$
\begin{equation*}
\begin{gathered} \, 0\to H^0(H,\mu(K))\xrightarrow{\alpha}H^0(H,U(K))\to H^0(H,\overline U(K))\to 0, \\ 0\to H^1(H,\mu(K))\xrightarrow{\beta} H^1(H,U(K))\to H^1(H,\overline U(K))\to 0, \end{gathered}
\end{equation*}
\notag
$$
whence it follows the assertion of the proposition for $K$. The proof for $L$ is just the same. This proves Proposition 3.3. Let $F$ be an arbitrary algebraic number field, $F_v$ the completion of $F$ relative to some place $v$ over $\ell$ and $F_{v,\infty}$ the cyclotomic $\mathbb Z_\ell$-extension of $F_v$. Put $\Gamma_v=G(F_{v,\infty}/F_v)$. If the extension $F_{v,\infty}/F_v$ is purely ramified then by the local class field theory we have the canonical epimorphism $U^{(1)}(F_v)\to \Gamma_v$, whose kernel coincides with the group $\mathcal{U}(F_v)$ of universal norms from all groups $U^{(1)}(F_{v,n})$ in the extension $F_{v,\infty}/F_v$, that is, $\mathcal{U}(F_v)=\cap_n N_n(U^{(1)}(F_{v,n}))$, where $N_n$ means the norm map from $F_{v,n}$ into $F_v$. As in [2], § 3, we denote by $P(F)$ the kernel of the natural map $\chi_F\colon \!\! \prod_{v\in S}\Gamma_v\to \Gamma$, where $\Gamma=G(F_\infty/F)$, and $\Gamma_v$ inserts in $\Gamma$ as a decomposition subgroup of $v$. Thus, if all places over $\ell$ purely ramify in $F_\infty/F$ then there are exact sequences
$$
\begin{equation}
1\to P(F)\to \prod_{v\in S}\Gamma_v\xrightarrow{\chi_F}\Gamma\to 1,
\end{equation}
\tag{3.10}
$$
$$
\begin{equation}
1\to \prod_{v\in S}\mathcal{U}(F_v)\to \prod_{v\in S}U^{(1)}(F_v)\xrightarrow{\pi_F}\prod_{v\in S}\Gamma_v\to 1.
\end{equation}
\tag{3.11}
$$
If $F$ is normal over $\mathbb Q$ then all groups entering (3.10) and (3.11) are $G(F/\mathbb Q)$-modules. The diagonal inclusion $U(F)[\ell]\hookrightarrow P(F)$ combined with (3.11) induce a map $\varphi_F\colon U(F)[\ell]\to P(F)$, whose kernel, which we denote by $U_2(F)$, – is the subgroup of all local universal norms from $F_\infty$ in the group $U(F)[\ell]$. In particular, in the cases $F=K$ or $F=L$ we have the maps $\varphi_K$, $\varphi_L$ and the groups $U_2(K)$, $U_2(L)$. Proposition 3.4. The maps $\varphi_K$ and $\varphi_L$ are epimorphisms. In other words, there are exact sequences Proof. For $K$ the desired assertion was already proved in [2], Proposition 4.2. For $L$ the proof is even simpler. Let $M$ be the maximal Abelian $\ell$-extension of $L$ unramified out of $\ell$, and such that for any $v\in S$ the field $M_v$ contains in the cyclotomic $\mathbb Z_\ell$-extension of $L_w$, where $w$ is a place of $L$ under $v$. Then, taking into account, that $\operatorname{Cl}(L)_\ell=0$ by Proposition 3.1 and applying the class field theory, we get that $G(M/L_\infty)\cong P(L)/\operatorname{Im}\varphi_L$, but according to Proposition 3.1 we have $\overline T_\ell(L_\infty)=0$, whence it follows $M=L_\infty$. This proves Proposition 3.4. In each of the extensions $K/k$ and $L/k$ the place $v\in S$ of $K$ splits completely, hence (3.10) yields where $I_H$ and $I_{\widetilde H}$ are the ideals of augmentation of the group rings $\mathbb Z_\ell[H]$ and $\mathbb Z_\ell[\widetilde H]$ respectively. In particular, $H^0(H,P(K))=0$ and $H^{-1}(H,P(K))\cong \mathbb Z_\ell/\ell \mathbb Z_\ell$. The analogous result holds for $P(L)$.We put $\overline U_2(K)=U_2(K)/\mu_\ell(K)$ and $\overline U_2(L)=U_2(L)/\mu_\ell(L)$. Proposition 3.5. Let $\mathfrak P_1$ be a prime divisor in $K$ (or in $L$) of the divisor $\mathfrak p_1=(p_1)$, where $p_1$ is a prime number entering in $a$ in (3.1) (or in $b$ in (3.2)). Let the order of $\mathfrak P_1$ in the group $\operatorname{Cl}(K)$ (respectively, in the group $\operatorname{Cl}(L))$ is prime to $\ell$. (Note that the last condition always holds for $L$ because of Proposition 3.1.) Then the following equalities hold Proof. Let $x\in K^\times$ be such an element that $(x)=\mathfrak P_1^h$ for some $h$ prime to $\ell$. The element $x$ is defined up to multiplication by an arbitrary unit in $U(K)$. Then $x$, as well as $p_1$, is a unit in the local field $K_v$ for any $v$ over $\ell$. Therefore, under the diagonal inclusion $K^\times\hookrightarrow \prod_{v\in S}K_v^\times$ the elements $x$ and $p_1$ go into $\prod_{v\in S} U(K_v)$, so, we can consider they as elements of the group $\prod_{v\in S}U(K_v)[\ell]=\prod_{v\in S}U^{(1)}(K_v)$. Obviously, $N_{K/k}(x)=p_1^hu$, where $u$ is a unit in $U(K)$. Let $x_1=\pi_k(x)$, where $\pi_k$ is the map from (3.11) for $K$.
Since $\mathfrak P_1^\ell=\mathfrak p_1=(p_1)$ we obtain that in the group $\Gamma$ in (3.10) holds the relation $\chi_K(\pi_K(x)^\ell))=\chi_K(\pi_K(p_1^h))$, but $\chi_K(\pi_K(p_1^h))$ generates the group $\Gamma^\ell$. Hence the element $y=\chi_K(\pi_K(x))$ generates the group $\Gamma$. If $y'$ is some lifting of $y$ in $\prod_{v\in S}\Gamma_v$ then $y'$ generates this group as an $H$-module. Let $z$ be the image of $(\sigma-1)(x)$ in $\overline U(K)$. Then $z$ defines some element of the group $H^{-1}(H,\overline U(K))$, moreover, $\pi_K(z)$ generates the group $H^{-1}(H,P(K))\cong H^1(H,P(K))$. Therefore, the natural map $H^{-1}(H,\overline U(K)[\ell])\to H^{-1}(H,P(K))$, induced by the map $\varphi_K$ of (3.11), is an epimorphism. Then the exact cohomological sequence for (3.12) yields exact sequences
$$
\begin{equation*}
\begin{gathered} \, 0=H^{-2}(H,P(K))\to H^{-1}(H,\overline U_2(K))\to H^{-1}(H,\overline U(K)[\ell]\to H^{-1}(H,P(K))\,{\to}\, 0, \\ 0\to H^0(H,\overline U_2(K))\to H^0(H,\overline U(K)[\ell])\to H^0(H,P(K))=0. \end{gathered}
\end{equation*}
\notag
$$
This proves the proposition for $K$. The proof for $L$ is quite analogous (with using the exact sequence (3.13)). This proves Proposition 3.5. Theorem 3.1. There are two possibilities for $K$. (A) All three divisors $\mathfrak{P}_1$, $\mathfrak{P}_2$, $\mathfrak{P}_3$ have an order prime to $\ell$ in the group $\operatorname{Cl}(K)$. In this case
$$
\begin{equation*}
|H^0(H,\overline U_2(K))|=|H^1(H,\overline U_2(K))|=\ell.
\end{equation*}
\notag
$$
This case always take place if $|{\operatorname{Cl}_\ell(K)}|<\ell^{\ell-1}$. (B) At least, one of the divisors $\mathfrak{P}_1$, $\mathfrak{P}_2$, $\mathfrak{P}_3$ present non-zero element in $\operatorname{Cl}(K)_\ell$. Then For the field $L$ we always have Proof. Let $\mathfrak A=\mathfrak P_1^{m_1}\mathfrak P_2^{m_2}\mathfrak P_3^{m_3}$ be a product of prime divisors, which ramify in $K/k$. If $\mathfrak A$ is a principal divisor then by (3.3) the element $\mathfrak A\in (D_K^0)^H$ determines some element $\eta(\mathfrak A)$ in the group $H^1(H,U(K))$. Put
$$
\begin{equation*}
\mathcal{A} = (\mathbb Z\mathfrak P_1\oplus \mathbb Z\mathfrak P_2\oplus\mathbb Z\mathfrak P_3)/(\mathbb Z\mathfrak p_1\oplus\mathbb Z\mathfrak p_2\oplus\mathbb Z\mathfrak p_3)\cong (\mathbb Z/\ell\mathbb Z)^3.
\end{equation*}
\notag
$$
Then there is an exact sequence
$$
\begin{equation}
0\to (D_K^0)^H/D_k\to \mathcal{A}\xrightarrow{\psi}\operatorname{Cl}(K)_\ell^H.
\end{equation}
\tag{3.15}
$$
It follows from [1], Theorem 4.1, that $\operatorname{Cl}(K)_\ell$ is a cyclic $H$-module, that vanishes under the norm map $N_H$, therefore, by [1], Lemma 3.1, the group $\operatorname{Cl}(K)_\ell^H$ is of order $\ell$.
So, it follows from (3.15) that the group $H^1(H,U(K))=(D_K^0)^H/D_k$ is of order $\ell^2$ if $\psi\neq 0$, or of order $\ell^3$ if $\psi=0$. In the case (A) we have $\psi=0$, that is, $|H^1(H,U(K))|=\ell^3$. Calculating the Herbrand index $h(U(K))$ via Dirichlet theorem, we obtain
$$
\begin{equation*}
h(U(K))=|H^0(H,U(K))|/|H^1(H,U(K))|=\ell^{-1}, \text{ that is, } |H^0(H,U(K))|=\ell^2.
\end{equation*}
\notag
$$
Then by Proposition 3.2 we get $|H^1(H,\overline U(K))|=\ell^2$ and $|H^0(H,\overline U(K))|=\ell$. The condition $\psi=0$ means that the assumptions of Proposition 3.5 hold, that is,
$$
\begin{equation*}
|H^1(H,U_2(K))|=\ell \quad\text{and}\quad |H^0(H,U_2(K))|=\ell.
\end{equation*}
\notag
$$
Consider the lower central series of the $H$-module $\operatorname{Cl}(K)_\ell$. The first factor of this series is isomorphic to $\mathbb F_\ell(1)$ as a $\Delta$-module. Then [1], Lemma 3.2, yields that in the case $|{\operatorname{Cl}(K)_\ell}|<\ell^{\ell-1}$ this central series has no factors isomorphic to $\mathbb F_\ell(0)$ as a $\Delta$-module. This is the situation of the case (A).
Now we turn to the case (B). If one of the divisors $\mathfrak P_i$, $\mathfrak P_1$ for example, presents a non-zero element of $\operatorname{Cl}(K)$ then this element is fixed under the action of $G$. Then, applying Lemma 3.2 of [1] again and taking into account that $\operatorname{Cl}(K)_\ell/(\sigma- 1)\operatorname{Cl}(K)_\ell\cong \mathbb F_\ell(1)$ as a $\Delta$-module, we obtain that the length of the lower central series of the $H$-module $\operatorname{Cl}(K)_\ell$ is at least $\ell-1$, but its length cannot be bigger, since the group $\operatorname{Cl}(K)_\ell$ is of period $\ell$ by Corollary 3.1 of [2]. Therefore, in this case we have $|{\operatorname{Cl}(K)_\ell}|=\ell^{\ell-1}$. Since $\psi\neq 0$ in (3.15) we get
$$
\begin{equation*}
|H^1(H,U(K))|=\ell^2\quad\text{and}\quad |H^0(H,U(K))|=\ell.
\end{equation*}
\notag
$$
Now, it follows from Proposition 3.2 that $|H^1(H, \overline U(K))|\,{=}\,\ell$ and $|H^0(H,\overline U(K))|\,{=}\, 1$. This completes the proof of the theorem for $K$.
The proof for $L$ is analogous. We can write down an analogue of Formula (3.15) for the field $L$ again, but in this case we have $\operatorname{Cl}(L)_\ell^H=0$ by Proposition 3.1, and $\mathcal{A}\cong (\mathbb Z/\ell\mathbb Z)^2$, so, $|H^1(\widetilde H,U(L))|=\ell^2$ and $|H^0(\widetilde H,U(L))|=\ell$. By virtue of Proposition 3.2 we have $|H^1(\widetilde H,\overline U(L))|=\ell$ and $|H^0(\widetilde H,\overline U(L))|=1$. Then Proposition 3.5 yields that $H^1(\widetilde H, U_2(L))=H^0(\widetilde H, U_2(L))=0$. This proves Theorem 3.1. § 4. Structure of certain groups of units of the field $L$Let $\mathcal{A}(L)=\prod_{v\in S}U^{(1)}(L_v)$ and $\overline{\mathcal{A}}(L)=\prod_{v\in S}\overline U^{\,(1)}(L_v)$ be the maximal $\mathbb Z_\ell$-free factor module of $\mathcal{A}(L)$. Our nearest goal is to characterize $\mathcal{A}(L)$ as a $\mathbb Z_\ell[\widetilde G]$-module. To do this, we need some general results on $\mathbb Z_\ell[\widetilde G]$-modules, which are analogous to those in [1], § 3. Since the groups $G=G(K/\mathbb Q)$ and $\widetilde G=G(L/\mathbb Q)$ are isomorphic we shall formulate these general results for the case of $G$-modules. We shall consider a Noetherian $G$-module $A$ that is free as a $\mathbb Z_\ell[H]$-module. These conditions on $A$ are equivalent to simultaneous fulfilment of the following two conditions: (i) $A$ is Noetherian and has no torsion over $\mathbb Z_\ell$; (ii) $A$ is a cohomologically trivial $\mathbb Z_\ell[H]$-module. As in [1], we assume that some section $f\colon \Delta \to G$ is fixed. Let $\delta'=f(\delta)$ be a fixed generator of $f(\Delta)$ and $e_i$ the idempotent of $\mathbb Z_\ell[f(\Delta)]$ corresponding to the character $\omega^i$, where $\omega$ is the Teichmüller character. For $a\in A$ we denote by $a[i]$ the element $e_ia$. Let $\sigma$ be a fixed generator of $H$. As in [1], let $F_1=\mathbb Z_\ell[G]$ be a free $G$-module of rank $1$. Then we have
$$
\begin{equation*}
F_1/(\ell,(\sigma- 1))F_1\cong\bigoplus_{i=0}^{\ell-2}\mathbb F_\ell(i).
\end{equation*}
\notag
$$
If $a\in F_1$ is a generator of $G$-module $F_1$ then $a=\sum_{i-0}^{\ell-2}a[i]$ and thus
$$
\begin{equation}
F_1=Q(0)+Q(1)+\dots +Q(\ell-2), \ \text{ where } Q(i)=\mathbb Z_\ell[H]a[i].
\end{equation}
\tag{4.1}
$$
Any of $Q(i)$ is a $G$-module and has the rank at most $\ell$ as a $\mathbb Z_\ell$-module. Then it follows from (4.1) that the rank is exactly $\ell$, and (4.1) defines a decomposition of $F_1$ into a direct sum of $G$-modules $Q(i)$ for $i=0,1,\dots,\ell-2$. Therefore, each of the modules $Q(i)$ is projective as a $G$-module. Proposition 4.1. Let $A$ be a Noetherian $G$-module, which is free as a $\mathbb Z_\ell[H]$-module. Then there is a (non-canonical) isomorphism of $G$-modules where the exponents $r_i$ are defined uniquely by the module $A$. In particular, if $A$ is a cyclic $H$-module then $A\cong Q(i)$ for some $i$. Corollary 4.1. Suppose that we have an exact sequence of Noetherian $\mathbb Z_\ell[G]$-modules such that $A,B,C$ are free as $\mathbb Z_\ell$-modules. If any two modules of this sequence are projective then so is the third one. So, if $A$ is a free $\mathbb Z_\ell[H]$-module of rank $1$ then by Proposition 4.1we have $A\cong Q(i)$ for some $i$. In this case $A/(\ell,(\sigma -1))A\cong \mathbb F_\ell(i)$. In general, if $A/(\ell,(\sigma-1))A\cong \mathbb F_\ell(i)$ then we say that $A$ begins at $\mathbb F_\ell(i)$. Under the same assumption, if $A$ is finite and $A^H\cong \mathbb F_\ell(k)$ then we say that $A$ ends at $\mathbb F_\ell(k)$. Put $A\cong Q(i)$ and $B=A/\ell A$. Then $B$ is a cohomologically trivial $\mathbb F_\ell[H]$-module, and there is a lower central series
$$
\begin{equation*}
B=B_0\supset B_1\dots \supset B_\ell=0, \quad \text{where}\quad B_j/B_{j+1}\cong \mathbb F_\ell(i+j)
\end{equation*}
\notag
$$
for $j=0,1,\dots,\ell-1$, as it follows easily from Lemma 3.2 of [1]. The next Proposition is an analogue of Proposition 4.1 for the case $\mathbb F_\ell[G]$-modules and may be proved by the same arguments. Proposition 4.2. Let $\mathscr A$ be a Noether $G$-module, which is free as a $\mathbb F_\ell[H]$-module. Then there is a (non-canonical) isomorphism of $G$-modules where $\mathscr Q(i)=Q(i)/\ell Q(i)$ and the exponents $r_i$ are defined uniquely by $\mathscr A$. In particular, if $\mathscr A$ is a cyclic $H$-module then $\mathscr A\cong \mathscr Q(i)$ for some $i$. Corollary 4.2. Let an exact sequence of $\mathbb F_\ell[G]$-modules be given If any two modules of this sequence are projective in the category of Noetherian $\mathbb F_\ell[G]$-modules then so is the third one. Proposition 4.3. Let an exact sequence of Noetherian $\mathbb Z_\ell[G]$-modules be given and these modules are free as $\mathbb Z_\ell$-modules. If the $G$-module $A$ is projective then there is a decomposition into a direct sum $B\cong A'\oplus C'$, where $A\cong A'$ and $C\cong C'$. Proof. Put $A^0=\operatorname{Hom}_{\mathbb Z_\ell}(A,\mathbb Z_\ell)$ and let $B^0$ and $C^0$ be of analogous meaning. Let $A^0$, $B^0$ and $C^0$ has a standard structure of left $G$-modules, as it is explained in [3]. Then there is an exact sequence of $G$-modules with projective $A^0$. Therefore, it splits, that is, $B^0\cong C^0\oplus \widetilde A^0$, where $\widetilde A^0\cong A^0$. Thus $B\cong (B^0)^0 \cong(C^0\oplus \widetilde A^0)^0\cong C^{00}\oplus \widetilde A^{00}\cong C\oplus \widetilde A\cong A\oplus C$. This proves Proposition 4.3. Let $M(L)$ be the kernel of the natural map $\mathcal{A}(L)\to\overline{\mathcal{A}}(L)$. Hence, $M(L)=\prod_{v\in S}\mu_\ell(L_v)$. Proposition 4.4. The module $M(L)$ is isomorphic to $(\mathbb Z/\ell\mathbb Z)^\ell$ as an Abelian group. It is cyclic and cohomologically trivial as a $\widetilde H$-module, where $\widetilde H=G(L/k)$. The module $M(L)$ begins at $\mathbb F_\ell(1)$ and ends at $\mathbb F_\ell(1)$. So, taking into account the notions introduced above, we get an isomorphism of $\widetilde G$-modules $M(L)\cong \mathscr Q(1)$. Proof. The first assertion is obvious. The group $\widetilde H$ substitutes the factor $\mu_\ell(L_v)$, hence $M(L)$ is an induced $\widetilde H$-module. Therefore, it is cohomologically trivial. The norm map $N_{L/k}$ is a $\Delta$-homomorphism that maps $M(L)$ onto $M(k)=\mu_\ell(k)\cong \mathbb F_\ell(1)$. Therefore, $M(L)$ begins at $\mathbb F_\ell(1)$. This proves Proposition 4.4. We see that the $\widetilde G$-module $\overline{\mathcal{A}}(L)$ is induced as a $\widetilde H$-module. Therefore it is cohomologically trivial as a $\widetilde H$-module, and by Proposition 4.1 it has a decomposition of the form (4.2). Note that every direct summand $Q(i)$ enters this decomposition with multiplicity $1$, since $N_{L/k}$ induces an isomorphism
$$
\begin{equation}
\overline{\mathcal{A}}(L)/(\widetilde\sigma-1)\overline{\mathcal{A}}(L)\cong \overline{\mathcal{A}}(k)\cong \bigoplus_{i=0}^{\ell-2}\mathbb Z_\ell(i),
\end{equation}
\tag{4.4}
$$
where $\widetilde\sigma$ is a generator of $\widetilde H$ and $\mathbb Z_\ell(i)$ denotes a $\widetilde G$-module, which is isomorphic to $\mathbb Z_\ell$ as an Abelian group with trivial action of $\widetilde H$, while $\Delta$ acts on it by multiplication by $\omega^i$. In sequel we need the decomposition of $\mathcal{A}(L)$ with some additional properties. Proposition 4.5. $\widetilde G$-module $\mathcal{A}(L)$ has a decomposition into the direct sum of $\widetilde G$-modules $\mathcal{A}(L)=\bigoplus_{i=0}^{\ell-2}\mathcal{A}[i]$, where $\mathcal{A}[i]\cong Q(i)$ as a $\widetilde G$-module, and the following conditions hold: 1) $\mathcal{A}[2i]\subset U_2(L)$ for $i=1,\dots,(\ell-3)/2$; 2) the direct summand $\mathcal{A}[0]$ has a generator $\mathbf{a}_0$ such that $\Delta$ acts trivially on $\mathbf{a}_0$ and $(\widetilde\sigma-1)\mathbf{a}_0\in \overline U(L)$. Proof. The cohomological triviality of $\overline U_2(L)$ (see Theorem of 3.1) yields that $\overline U_2(L)/(\widetilde\sigma-1)\overline U_2(L)\cong \overline U(k)$, hence
$$
\begin{equation}
\overline U_2(L)\cong \bigoplus_{i=1}^{(\ell-3)/2} Q(2i).
\end{equation}
\tag{4.5}
$$
Then $\widetilde G$-module $\overline{\mathcal{A}}(L)/\overline U_2(L)$ is projective by Corollary 4.1, hence there is a decomposition into the direct sum $\overline{\mathcal{A}}(L)\cong (\overline{\mathcal{A}}(L)/\overline U_2(L))\oplus\overline U_2(L)$. Then we put $\overline{\mathcal{A}}[2i]=Q(2i)$, where $Q(2i)$ are the components of $\overline U_2(L)$ in the decomposition (4.5).
Let $\overline{\mathcal{A}}_2(L)$ be the subgroup of local universal norms (of $L_\infty$) in the group $\overline{\mathcal{A}}(L)$, so, that there is a natural isomorphism
$$
\begin{equation}
\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)\cong\prod_{v\in S}\Gamma_v,
\end{equation}
\tag{4.6}
$$
where $\prod_{v\in S}\Gamma_v$ has the same meaning, as in (3.9) and(3.10). Thus $\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)$ is a cohomologically trivial $\widetilde H$-module of $\mathbb Z_\ell$ rank $\ell$. This module admits a natural $\widetilde G$-epimorphism $\varphi\colon\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)\to \Gamma=G(L_\infty/L)$. Therefore by Proposition 4.2 we have $\overline{\mathcal{A}}(L)/\overline{\mathcal{A}}_2(L)\cong Q(0)$.
Now consider a $\widetilde G$-module $\overline{\mathcal{A}}(L)/\overline U_2(L)$. By Proposition 4.2 and formulae (4.3), (4.5) we have an isomorphism $B:=\mathcal{A}(L)/\overline U_2(L)\cong\bigoplus_{i=0}^{(\ell-3)/2} Q(2i+1)\oplus Q(0)$. Let $C=\mathcal{A}_2(L)/U_2(L)$ and $P(L)$ be of the same meaning as in (3.12). Then we can consider $C$ and $P(L)$ as submodules of $B$, where $C\cong \bigoplus_{i=0}^{(\ell-3)}Q(2i+1)$. Taking into account that $C$ and $P(L)$ have zero intersection in $B$, we obtain an exact sequence with a projective $\widetilde G$-module $C$, whence, taking factor-module with respect to $P(L)$, we get an exact sequence with projective $C$. Applying Proposition 4.3 to this sequence, we see that $B/P(L)$ contains a submodule $\Gamma'$, which is isomorphic to $\Gamma$ as a $\widetilde G$-module, that is, $\Gamma'$ is isomorphic to $\mathbb Z_\ell$ as an Abelian group and $\widetilde G$ acts trivially on $\Gamma'$. Thus, we have an exact sequence where $C'\cong C$. Let $a'$ be a generator of $\Gamma'$. Then, lifting $a'$ up to some element $a''\in B$, we obtain an element $a''\in B$ such that $b:=(\widetilde \sigma-1)a''\in P(L)$. If a $\widetilde H$-submodule generated by $b$ would be a proper submodule of $P(L)$ then , after replacing $b$ by some $b'=b+x$ for a suitable $x\in P(L)$, we should obtain $\widetilde\sigma (b')=b'$. It means that $B\cong C\oplus P(L)\oplus \mathbb Z_\ell b'$ as an $\widetilde H$-module. But this contradicts to projectivity of $B$.Finally, we put $\mathbf{a}=a''[0]$. Obviously, $a''$ and $\mathbf{a}$ are equal modulo $P(L)$, $\mathbf{a}$ generates a submodule of $B$, which is isomorphic to $Q(0)$, and $\mathbf{a}$ is fixed under the action of $\Delta$. This completes the proof of Proposition 4.5. Using Proposition 4.5, we shall obtain some special decomposition for $\mathscr B(L):=\overline{\mathcal{A}}(L)/\ell\overline{\mathcal{A}}(L)$ that we need in the next section. We shall construct a decomposition into the direct sum $\mathscr B(L)=\bigoplus_{i=0}^{\ell-2}\mathscr B[i]$, where for $i\neq 1$ we put $\mathscr B[i]=\mathcal{A}[i]/\ell\mathcal{A}[i]$ and for $i=1$ we define the component $\mathscr B[1]$ as follows. Let $L_v$ be the completion of $L$ at the place $v$ over $\ell$. The group $U(L_v)$ contains $\ell$-primary elements, that is, such elements $u_v$, that $L_v(\sqrt[\ell]{u_v}\,)$ is an unramified extension of $L_v$ of degree $\ell$. We shall consider $u_v$ as an element of the group $\overline U(L_v)/(\overline U(L_v))^\ell$. Thus, there are $\ell-1$ primary elements, which together with the unity form a subgroup denoted by $\mathcal{P}_v$ of the group $\overline U(L_v)/\overline U(L_v)^\ell$. We put $\mathscr B[1]=\prod_{v\in S}\mathcal{P}_v$ and consider this group as a subgroup of $\mathscr B$. Obviously, $\mathscr B[1]$ is a Galois submodule in $\mathscr B(L)$. The group $\widetilde H$ substitutes the components $\mathcal{P}_v$ of $\mathscr B[1]$, so, $\mathscr B[1]\cong \mathbb F_\ell[\widetilde H]$ as an $\widetilde H$-module. Proof. Put $G_v^{\mathrm{un}}=G(L_v(\sqrt[\ell]{u_v}\,)/L_v)$. Then by the local class field theory there is a canonical isomorphism $G_v^{\mathrm{un}}\cong L^\times_v/(L_v^\times)^\ell U(L_v)$. Thus, if $D_S$ is the group of divisors of $L$ with supports in $S$ then $D:=\prod_{v\in S}G_v^{\mathrm{un}}\cong D_S/D_S^\ell$. This defines a structure of Galois module on $D$. Since there is an epimorphism of Galois modules $D\to \mathbb F_\ell(0)$ induced by the norm map, we get that $D\cong \mathscr B[0]$, that is $D$ starts with $\mathbb F_\ell(0)$ and ends with $\mathbb F_\ell(0)$. On the other hand, by Kummer theory there is a non-degenerate pairing between $D$ and $\mathscr B[1]$, hence $\mathscr B[1]$ starts with $\mathbb F_\ell(1)$ and ends with $\mathbb F_\ell(1)$. This proves Proposition 4.6. Proposition 4.7. The Galois module $\mathscr B(L)$ has a decomposition into the direct sum Proof. Put
$$
\begin{equation}
\mathscr B'(L)=\bigoplus_{i=0,\,i\neq 1}^{i=\ell-2}\mathscr B[i].
\end{equation}
\tag{4.9}
$$
To prove the formula (4.8) it is enough to check that in the group $\mathscr B(L)$ we have $C:=\mathscr B'(L)\cap\mathscr B[1]=0$. To do this, it is enough to check that $C^{\widetilde H}=0$. Obviously, $C^{\widetilde H}=\mathscr B'(L)^{\widetilde H}\cap \mathscr B[1]^{\widetilde H}$, but we have $\mathscr B'(L)^{\widetilde H}\cong \bigoplus_{i=0,\, i\neq 1}^{\ell-2}\mathscr B[i]^{\widetilde H}\cong \bigoplus_{i=0,\, i\neq 1}^{i=\ell-2}\mathbb F_\ell(i)$, while $\mathscr B[1]^{\widetilde H}\cong\mathbb F_\ell(1)$. This proves the existence of the decomposition (4.8). The main results of this section we can formulate as a following theorem. Theorem 4.1. For the field $L$, the Galois module $\mathcal{A}(L)/\mathcal{A}(L)^\ell$ contains in the split exact sequence of Galois modules This result may be presented in another form, taking into account that $M(L)$ and $\mathscr B[1]$ are defined canonically as submodules of $\mathcal{A}(L)/\mathcal{A}(L)^\ell$. Namely, there is a split exact sequence of Galois modules
$$
\begin{equation}
0\to \mathscr B[1]\oplus M(L)\to \mathcal{A}(L)/\mathcal{A}(L)^\ell \to \mathscr B'(L)\to 0,
\end{equation}
\tag{4.11}
$$
where $\mathscr B'(L)$ was defined in the proof of Proposition 4.7. § 5. Certain $\ell$-extensions of the field $L$In this section we assume that the field $L=k(\sqrt[\ell]{b})$ defined as in (3.2) is fixed. In particular, we fix the pair of primes $p_1,p_2\,{\in}\,\mathbb P_0$ such that $b=p_1^{s_1}p_2^{s_2}$. Let $p_3\in \mathbb P_0$, where $p_3$ has the same meaning as in (3.1). Let $N=N(p_3)$ be the maximal Abelian $\ell$-extension of $L$ such that the Galois group $G(N/L)$ is of period $\ell$, and only prime divisors of $p_3$ may ramify in the extension $N/L$. We wish to obtain the spectrum of all possible values of degree $[N:L]$. Note that we have always $[N:L]>1$ since $N\supseteq K\cdot L$. To avoid plenty of indices, we shall denote $p_3$ by $q$ in this section. By $(\mathfrak q)$ we denote the principal divisor of $q$, which is a prime divisor in $\mathbb Q$ or in $K$. Proof. If $\mathfrak q$ remains prime in $L/k$ then $\mathfrak q$ remains prime in $L/\mathbb Q$. It means that $\widetilde G=G(L/\mathbb Q)$ coincides with the decomposition subgroup of $\mathfrak q$. But $\mathfrak q$ does not ramify in $L/\mathbb Q$, hence its decomposition subgroup must be cyclic hence it cannot coincide with $\widetilde G$. This concludes the proof. Let, as in the beginning of § 3 some section $f\colon \Delta\to \widetilde G$ is fixed. Since there are exactly $\ell$ subgroups of order $\ell-1$ in $\widetilde G$ and all these subgroups are mutually conjugate, we may assume that the prime divisors $\mathfrak q_1,\dots,\mathfrak q_\ell$ are enumerated in such a way that $f(\Delta)$ coincides with the decomposition subgroup of $\mathfrak q_1$. Since the Galois group $G(N/L)$ is of period $\ell$ we can use the Kummer theory for description of $N$. Suppose that $L(\sqrt[\ell]{\alpha}\,)\subseteq N$. This means that all divisors prime to $q$ enter $\alpha$ with exponents divisible by $\ell$. Let $ h$ be the class number of $L$, which is prime to $\ell$ by Proposition 3.1, and $\bf h$ is such number that $\mathbf{h}\equiv 0\pmod h$ and $\mathbf{h}\equiv 1\pmod\ell$. Then, putting $\alpha'=\alpha^{\mathbf{h}}$, we get $L(\sqrt[\ell]{\alpha}\,)=L(\sqrt[\ell]{\alpha'}\,)$. If $\alpha$ contains a prime divisor distinct from $\mathfrak q_1,\dots,\mathfrak q_\ell$, the divisor $\mathfrak p^{r_\mathfrak p}$, for example, then $r_{\mathfrak p}\equiv 0\pmod\ell$ and $(\alpha')$ contains $\mathfrak p$ with exponent $r_\mathfrak p\mathbf{h}$. But the divisor $\mathfrak p^{\mathbf{h}}$ is principal, hence, multiplying $\alpha'$ by some $\ell$-th power if necessary, we may assume that $\alpha'$ contains only the prime divisors $\mathfrak q_1,\dots, \mathfrak q_\ell$. Thus, we have to characterize all the elements $\alpha'\in L$ such that $(\alpha')$ contains only prime divisors $\mathfrak q_1,\dots,\mathfrak q_\ell$ while all the places $v\in S$ split or remain unramified in $L(\sqrt[\ell]{\alpha'}\,)/L$. Proposition 5.2. Let $\alpha_1,\alpha_2\in L$ be such elements that $L_1:=L(\sqrt[\ell]{\alpha_1}\,)\subseteq N$ and $L_2:=L(\sqrt[\ell]{\alpha_2}\,)\subseteq N$. Then the equality of the divisors $(\alpha_1)=(\alpha_2)$ yields that $L_1=L_2$ and $\alpha_1=\alpha_2u^\ell$, where $u$ is a unit of $L$. Proof. Put $u_1=\alpha_1\alpha_2^{-1}$. Then $u_1$ is a unit of $L$ and $L(\sqrt[\ell]{u_1}\,)\subseteq N$. The extension $L(\sqrt[\ell]{u_1}\,)/L$ cannot have ramification out of $S$, but it cannot have ramification in $S$ by definition of $N$. Therefore, the extension $L(\sqrt[\ell]{u_1}\,)/L$ is unramified, but then it follows from Proposition 3.1 that $L(\sqrt[\ell]{u_1}\,)=L$, that is, $u_1=u^\ell$ for some unit $u\in U(L)$. This proves Proposition 5.2. Let $\beta_1\in L^\times$ be such an element that $(\beta_1)=\mathfrak q_1^{\mathbf{h}}$. In this case the element $u:=\beta^{1-\delta}$ is a unit. Since $H^{-1}(\Delta,U(L)[\ell])=0$ we get that there are units $u_1$ and $u_2$ such that $u=u_1^{1-\delta}u_2^\ell$. Then, putting $\beta=\beta_1u^{-1}$, we obtain such an element $\beta$ that
$$
\begin{equation}
(\beta)=\mathfrak q_1^{\mathbf{h}} \text{ and } \beta^{1-\delta}=u_2^\ell\quad \text{for some } u_2\in U(L)[\ell].
\end{equation}
\tag{5.1}
$$
Any divisor with support in $\mathfrak q_1,\dots,\mathfrak q_\ell$ may be written (in additive notation) in the form $\eta(\beta)$ for some $\eta\in \mathbb Z_\ell[\widetilde H]$. The element $\eta\beta$ is a unit in the local field $L_v$ for all $v\in S$, so, we can consider $\eta\beta$ as an element of $\mathcal{A}(L)$. By $\pi(\eta\beta)$ we denote the image of $\eta\beta$ in $\mathcal{A}(L)/\ell\mathcal{A}(L)$. Using the exact sequence (4.11), we can transform $\pi(\eta\beta)$ into the element $\pi'(\eta\beta)\in\mathscr B'(L)$. Then it follows from (4.9) that $\pi'(\eta\beta)=\bigoplus_{i=0,\,i\neq 1}^{i=\ell-2}\pi_i(\eta\beta)$, where $\pi_i(\eta\beta)\in\mathscr B[i]$. Formula 4.5 and the definition of the component $\mathscr B[i]$ yield that
$$
\begin{equation*}
\bigoplus_{i=1}^{(\ell-3)/2}\mathscr B[2i]\cong \overline U_2(L)/\ell\overline U_2(L),
\end{equation*}
\notag
$$
so, after multiplying $\beta$ by some suitable unit in the group $\overline U_2(L)$, we may assume and we shall assume in further that $\pi_i(\beta)=0$ for $i=2,4,\dots,\ell\,{-}\,3$. Then the same property have the elements $\pi(\eta\beta)$ for any $\eta$ and $i=2,4,\dots,\ell-3$. So, we can assume that some element $\beta\in L^\times$ is chosen such that $\pi(\beta)^\delta=\pi(\beta)$, moreover, $\pi_i(\beta)=0$ for $i=2,4,\dots,\ell-3$. We shall consider a system $\eta_1,\dots,\eta_{\ell^\ell}\in \mathbb Z_\ell[\widetilde H]$ that presents all the elements of $\mathbb F_\ell[\widetilde H]$, and we have to determine all $\eta_j$ such that $L(\sqrt[\ell]{\eta_j\beta}\,)\subseteq N$. The elements $\eta_j\beta$ must satisfy the following condition
$$
\begin{equation}
\pi_i(\eta_j\beta)=0 \text{ for } i=2,\dots,\ell-2\quad\text{and}\quad \pi_M(\eta_j\beta)=0,
\end{equation}
\tag{5.2}
$$
where the projection $\pi_M\colon\mathcal{A}(L)/\ell\mathcal{A}(L)\to M(L)$ is defined by the split exact sequence (4.10). Herein we can assume that the conditions (5.2) hold already for even $i>0$. The conditions that appear in the case $i=0$, we shall discuss below (see Proposition 5.3). Obviously, the images of those representatives of $\eta_1,\dots,\eta_{\ell^\ell}$ that satisfy the condition (5.2) form an ideal $I$ of $\mathbb F_\ell[\widetilde H]$, and $I=I(q)$ is of the form $I=I_{\mathrm{max}}^i$, where $I_{\mathrm{max}}$ is the maximal ideal of $\mathbb F_\ell[\widetilde H]$. Proof. Indeed, there is always the extension $KL/L$, that corresponds to $\eta=\sum_{h\in \widetilde H}h$, hence $I$ always contains the ideal $\mathbb F_\ell(\sum_{h\in \widetilde H}h)$. Proposition 5.4. For the field $L$ we have always a strict inclusion $I\subset \mathbb F_\ell[\widetilde H]$. Proof. We suppose that $(q)$ remains prime in the extension $k_\infty/k$. This means that the Artin symbol of divisor $(q)$ is a generator of $\Gamma=G(k_\infty/k)$. The map of restriction of automorphisms from $L_\infty$ to $k_\infty$ maps isomorphically $\Gamma_1:=G(L_\infty/L)$ onto $\Gamma$. This map agree with the norm map $N_{L/k}$, which sends $\mathfrak q_1$ into $(q)$. It means that for any $\beta\in L^\times$ such that $(\beta)=\mathfrak q_1^{\mathbf{h}}$, there is a place $v\in S$ of $L$ such that the inclusion $v\colon L\hookrightarrow L_v$ sends the element $\beta$ into a generator of $\Gamma_v$, where $\Gamma_v$ is of the same meaning as in (3.10). It is known by local class field theory that the extension $L(\sqrt[\ell]{\beta}\,)/L$ is ramified for such $\beta$, hence, for example, $1\notin I$. This proves Proposition 5.4. Remark. Consider an element $\eta\beta$ such that $\eta\in I_{\mathrm{max}}$. Then $\pi_0(\eta\beta)$ belongs to $\pi_0(P(L))$, where $P(L)$ has the same meaning as in (3.10). This means that, multiplying $\beta$ by some unit in $P(L)$, we always may assume that $\pi_0(\eta\beta)=0$ (under assumption that $\eta\in I_{\mathrm{max}}$). Now we consider the restrictions that yields the equality $\pi_i(\eta\beta)=0$ for some odd $i=3,5,\dots,\ell-2$. Remind that we use additive notation for multiplication. The lower central series of the Galois module $\mathscr B[i]$ starts with $\mathbb F_\ell(i)$. Meanwhile, the element $\beta$ is fixed under the action of $\Delta$. If $\pi_i(\beta)=0$, then there appear no obstructions connected with the projection $\pi_i$. If $\pi_i(\beta)\neq 0$ then, applying Lemma 3.2 of [1] in this case, we obtain that $\pi_i$ sends $\beta$ into a generator of $\mathscr B[i]^{(\widetilde\sigma-1)^{\ell-1-i}}$ ($\Delta$ acts identically on $\beta$ and the module $\mathscr B[i]^{(\widetilde\sigma-1)^{\ell-1-i}}$ starts with $\mathbb F_\ell(0)$). Then the element $\pi_i(\beta)$ generates in $\mathscr B[i]$ a submodule of order $\ell^{i+1}$. Thus, in this case we have $\pi_i(\eta\beta)=0$ if and only if $\eta\in I_{\mathrm{max}}^{i+1}$. We have to consider two particular cases else: the behavior of $\pi_1(\beta)$ and the behavior of the element $\pi_M(\beta)$. In the case $i=1$ we, repeating all preceding considerations, obtain that $\pi_1(\beta)$ generates a submodule of order $\ell^2$ in $\mathscr B[1]$. But since $I$ is a proper ideal, we have either $\pi_1(\eta\beta)=0$, which corresponds to the extension $L(\sqrt[\ell]{\eta\beta}\,)/L$, where all places over $\ell$ completely split, or $\pi_1(\eta\beta)\neq 0$, and this corresponds to the case when all places over $\ell$ of the field $L$ remain unramified in the extension $L(\sqrt[\ell]{\eta\beta}\,)/L$. At last, let us consider $\pi_M(\beta)$. Since $M$ starts by $\mathbb F_\ell(1)$ and ends by $\mathbb F_\ell(1)$, we obtain, as in the case $i=1$, that $\pi_M(\eta\beta)\in M^{\widetilde H}=\mu_\ell(L)$, hence, after multiplying the element $\eta\beta$ by a suitable root of unity in $\mu_\ell(L)$, we can assume that $\pi_M(\eta\beta)=0$. We can formulate the results of our study in the form of the following theorem. Theorem 5.1. Let $X:=3,5,\dots, \ell-2$ and $X_1$ be the subset of those $i\in X$, for which $\pi_i(\beta)\neq 0$. Let $i_0$ be the maximal index in $X_1$. Then If $X_1$ is empty then $I(q)=I_{\mathrm{max}}$. Here either all places over $\ell$ split completely in $N/L$, or each of them has inertia degree $\ell$. Proof. Any odd index $i>1$, that is, any $i\in X_1$ must satisfy the condition $I(q)\subseteq I_{\mathrm{max}}^{i+1}$, whence it follows (5.3). If $X_1$ is empty then any element $\eta\beta$ is admissible, that is $I(q)=I_{\mathrm{max}}$. If $\pi_1(\eta\beta)=0$ for any $\eta$ then all places over $\ell$ split in the extension $N/L$. Otherwise, all places over $\ell$ have non-unit inertia degree, which, as man can check easily is equal to $\ell$. This proves Theorem 5.1. Indeed, $|(I(q))|$ is the order of the Kummer group of the extension $N/L$, which coincides with $I(q)$. The index $i_0$ is odd by definition, $|(I_{\mathrm{max}})|=\ell^{\ell-1}$, so it follows from (5.3) that $|I(q)|$ is an even power of $\ell$. Then the degree $[N:L]$ is equal to $\ell^{r(q)}$, where $r(q)=\ell-i_0-2$ is even. In the case $\ell=3$ the set $X_1$ is empty, therefore, $[N:L]=9$, and all the places over $\ell$ either split completely (if $\pi_1(\beta)= 0$), or have inertia degree $3$ (if $\pi_1(\beta)\neq 0$). Till now we studied how the properties of the field $N(q)$, in particular, its degree over $L$ depend on $q$. Now we shall treat the inverse problem: to prove the existence of prime $q$ such that the field $N(q)$ has a given properties. Namely, the following theorem holds. Theorem 5.2. Assume that for the set of indices $i=1,2,\dots,\ell-2$ a collection of elements $\alpha_i\in\mathscr B[i]$ is given such that $\alpha_i$ are fixed under the action of $f(\Delta)$. Then there are infinitely many prime divisors $(q)$ of $\mathbb Q$ such that $(q)$ remains prime in $k_\infty$, $(q)$ splits completely in $L/k$ and $(q)$ has the following additional property: Let $\mathfrak q_1$ be a prime divisor of $(q)$ in $L$ fixed by the action of $f(\Delta)$. Then there is $\beta=\beta(\mathfrak q_1)$ such that the principal divisor of $\beta$ is equal to $\mathfrak q_1$, and the following properties hold: 1) $\pi_0(\beta(\mathfrak q_1))\notin P(L)/\ell P(L)$; 2) $\pi_i(\beta(\mathfrak q_1)) =\alpha_i$ for $i=1,2,\dots,\ell-2$. Proof. Consider the finite Galois module $\mathscr B(L)=\mathcal{A}/\ell\mathcal{A}$. Let $\widetilde U(L)$ be an image of $U(L)/\ell U(L)$ in the group $\mathscr B(L)$. As it was shown in Propositions 4.5 and 4.6, in $\widetilde U(L)$ there is a subgroup $U_2(L)/\ell U_2(L)$ and there is a subgroup $P(L)/\ell P(L)$ that contains in the component $\mathscr B_0$.
According to the global class field theory, man can interpret the group $R(L):=\mathscr B(L)/\widetilde U(L)$ as a Galois group of some Abelian extension $F_1(L)/L$. Put $F(L)=F_1(L)^{M(L)}$. Then the element $\varphi:=\prod_{i=1}^{\ell-2}\alpha_i^{-1}$ may be considered as an element of the group $R(L)$, that is, as an automorphism $\varphi$ of $F(L)/L$. By the Chebotarev density theorem there are infinitely many prime divisors $\mathfrak q_1'$ of $L$ such that $\varphi$ coincides with Frobenius automorphism that corresponds to $\mathfrak q_1'$. The divisor $\mathfrak q_1'$, that we have constructed, has the desired projections $\alpha_i^{-1}$, but we have to check if the divisor $(q)$ under $\mathfrak q_1$ remains prime in $k_\infty$? This condition is equivalent to the claim that $(q)$ remains prime simultaneously in the extension $\mathbb Q_\infty/\mathbb Q$ and in $k/\mathbb Q$. The fact that $(q')$ remains prime in $\mathbb Q_\infty/\mathbb Q$ follows from the fact that $\pi_0(\beta(q'_1))$ does not contain in the group $P(L)/\ell P(L)\subset \mathscr B[0]$. We have already discussed this situation in detail during the proof of Proposition 5.4. Consider a tower of fields $F(L)\supset L\supset \mathbb Q$. The group $G(L/\mathbb Q)$ contains the subgroup $f(\Delta)$, which fixes the divisor $\mathfrak q'_1$. Since $G(L/\mathbb Q)$ acts on $F(L)$ by conjugation, we obtain that $\varphi\in G(F(L)/L)$ commutes with the generator $\widetilde\delta$ of $f(\Delta)$. Then the product of these two commuting elements $\varphi$ and $\widetilde\delta$ generates in $G(F(L)/\mathbb Q)$ a cyclic subgroup $C$ of order $\ell(\ell-1)$. By the Chebotarev density theorem for any $\mathfrak q'_1$ there are infinitely many prime divisors $\mathfrak Q_1$ of $F(L)$ such that the decomposition subgroup of $\mathfrak Q_1$ coincides with $C$. Let $\varphi_1$ be a Frobenius automorphism in $G(F(L)/\mathbb Q)$ that corresponds to $\mathfrak Q_1$. Without restricting the generality, we can assume that $\varphi_1^{\ell-1}=\varphi$. The divisor $\mathfrak Q_1$ remains prime in the extension $F(L)/F(L)^\varphi$, that is, $\mathfrak Q_1$ is a prime divisor of the field $F(L)^\varphi$. Let $\mathfrak q_1$ be a prime divisor of $L$ under $\mathfrak Q_1$. Then $N(\mathfrak Q_1)=\mathfrak q_1$, where $N$ is the norm in the extension $F(L)^\varphi/L$. So, for the element $\beta_1=\beta_1(\mathfrak q_1)$ such that the principal divisor $(\beta_1)$ is equal to $\mathfrak q_1$, we obtain that the projections $\pi_i(\beta^{\ell-1})$ are equal to $\alpha_i$, that is, $\mathfrak q_1^{\ell-1}$ is a suitable divisor. This proves Theorem 5.2. Corollary 5.2. For any pair of distinct prime numbers $p_1,p_2\in \mathbb P_0$ there are infinitely many primes $p_3=q\in \mathbb P_0$ such that $[N:L]=\ell^i$ for any even $i$, where $2\leqslant i\leqslant \ell-1$. Corollary 5.3. For any pair of distinct prime numbers $p_1,p_2\in \mathbb P_0$ there are infinitely many primes $p_3=q\in \mathbb P_0$ such that $[N:L]=\ell^{\ell-1}$, and all the places over $\ell$ split completely in the extension $N/L$, and infinitely many primes $p_3=q\in\mathbb P_0$ such that all the places over $\ell$ have inertia degree $\ell$ in the extension $N/L$. § 6. Application to arithmetic of $K$Proposition 6.1. Assume that $\ell > 3$, $p_3=q$ and $N=N(q)$ has the same meaning as in § 5. Then the following three conditions are equivalent: (i) $d(\operatorname{Cl}_S(K)_\ell)\geqslant 2$; (ii) $d(\operatorname{Cl}_S(LK)_\ell)\geqslant 1$; (iii) $d(G(N/L))\geqslant 2$. Proof. (i) $\Rightarrow$ (ii). Let $d(\operatorname{Cl}_S(K)_\ell)\geqslant 2$. This means that there is an unramified extension $N_1/K$ of $K$, with the Galois group $(\mathbb Z/\ell\mathbb Z)^2$ such that all places over $\ell$ split completely in this extension. Then $N_1L/LK$ is an unramified Abelian $\ell$-extension, and all places over $\ell$ split in it, that is, $d(\operatorname{Cl}_S(LK)_\ell)\geqslant 1$.
(ii) $\Rightarrow$ (iii). Let $N_2$ be the maximal Abelian unramified $\ell$-extension of $LK$, in which all places over $\ell$ completely split. Then the Galois group $G(KL/L)\cong \mathbb Z/\ell \mathbb Z$ acts by conjugation on the Abelian $\ell$-group $G(N_2/KL)$. Hence there is a non-trivial factorgroup $C=G(N_2/KL)/B$ with identical action of $G(KL/L)$. Put $N_3=N_2^B$. Then the Galois group $G(N_3/KL)$ contains in the exact sequence whence it follows that $G(N_3/L)$ is an Abelian $\ell$-group of order at least $\ell^2$. Only the places $\mathfrak q_1,\dots,\mathfrak q_\ell$ over $q$ may be ramified in the extension $N_3/L$, and $G(N_3/L)$ is generated by the inertia subgroups of these place. Each of these subgroups is of order $\ell$ or $1$. Therefore, $N_3\subseteq N$ and $G(N_3/L)$ is of period $\ell$, so $d(G(N/L))\geqslant d(G(N_3/L))\geqslant 2$.(iii) $\Rightarrow$ (i). Since $K\subseteq N$ the condition $d(G(N/L))\geqslant 2$ yields $[N:KL]\geqslant \ell$. Thus, $[N:K]\geqslant \ell^2$. Let $N_4$ be the maximal Abelian unramified extension of $KL$, such that all the places over $\ell$ split completely in it. Then, using the same method, that we have used above to construct the field $N_3$, and an analogue of the exact sequence (6.1), which in the present case has the form where $C'$ is the maximal factorgroup of $G(N_4/KL)$ with trivial action of $G(KL/K)$, we obtain an Abelian unramified $\ell$-extension $N_5/K$ of degree $\geqslant\ell^2$ such that all places over $\ell$ split completely in it. By Theorem 4.1 of [1] the $H$-module $G(N_5/K)$ is cyclic. So, by Theorem 3.1 of [1], if the period of $\operatorname{Cl}_S(K)_\ell$ is greater $\ell$, then $d(\operatorname{Cl}_S(K)_\ell)\geqslant \ell-1$. So, the condition $[N_5:K]\geqslant \ell^2$ yields $d({\rm CL}_S(K)_\ell)\geqslant 2$. This proves Proposition 6.1.The following theorem 6.1 is an immediate consequence of Corollary 5.2 and Proposition 6.1. Theorem 6.1. Assume that $\ell > 3$ and $K$ is of the form (3.1). In this case we have always $d(\operatorname{Cl}_S(K)_\ell)\geqslant 2$. In the case $\ell=3$ we need the following variant of Proposition 6.1. Proposition 6.2. Under conditions of Proposition 6.1 assume that $\ell\,{=}\,3$. Let $N_6$ be the maximal Abelian $\ell$-extension of $L$ unramified out of the prime divisors $\mathfrak q_1$, $\mathfrak q_2$, $\mathfrak q_3$, over $(q)=(p_3)$ and such that $G(N_6/L)$ is of period $\ell$. We assume additionally that all the places over $\ell$, must be unramified, but we do not assume that they split completely in $N_6/L$. Then the following three conditions are equivalent: (i) $d(\operatorname{Cl}(K)_\ell)\geqslant 2$; (ii) $d(\operatorname{Cl}(KL)_\ell)\geqslant 1$; (iii) $d(G(N_6/L))\geqslant 2$. The next theorem is an immediate consequence of Corollary 5.2, Theorem 6.1 and Proposition 6.2. Theorem 6.2. Let $\ell=3$ and $K$ be of the form (3.1). Then for any pair $p_1,p_2\in \mathbb P_0$ there are infinitely many $p_3\in \mathbb P_0$ such that $d(\operatorname{Cl}_S(K)_\ell)=1$ and infinitely many $p_3$ such that $d(\operatorname{Cl}_S(K)_\ell)>1$. 
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