$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Cl{Cl}$[Reference: S. Lang, Cyclotomic Fields I and II, §2 chap 6]
Let $K$ be a number field. Suppose that $K$ contains the $p$-th roots of unity if $p$ is odd and it contains $i$ if $p=2$. Let $q_0$ be the power of $p$ such that the $q_0$-th roots of unity lie in $K$. Let $q_n=q_0p^n$ and $K_n=K(\mu_{q_n})$. Denote by:
$K_\infty$ the cyclotomic $\mathbb{Z}_p$-extension of $K_0$;
$\Omega$ the maximal $p$-abelian $p$-ramified extension of $K_\infty$;
$E_n$ the units in $K_n$ and $E=\bigcup_n E_n$; $\Omega_E=K_\infty\bigl(E^{1/p^\infty}\bigr)$;
$A_n$ the group of elements $\alpha\in K_n^*$ such that $(\alpha)=\mathfrak{a}^{p^n}$, where $\mathfrak{a}$ is fractional ideal prime to $p$ and $A=\bigcup_n A_n$; $\Omega_A=\bigcup_n K_\infty\bigl(A_n^{1/p^n}\bigr)$;
$B_n$ the $p$-units in $K$, i.e. the group of elements whose ideal factorization consists in ideals dividing $p$; $\Omega_B=K_\infty\bigl(B^{1/p^\infty}\bigr)$.
The aim is to study the structure of the extensions $\Omega_A/\Omega_E$ and $\Omega_B/\Omega_E$. We have the following result.
Theorem The Galois groups $\Gal(\Omega_A/\Omega_E)$ and $\Gal(\Omega_B/\Omega_E)$ are $\Lambda$-torsion modules. Therefore $\Gal(\Omega/\Omega_E)$ is a $\Lambda$-torsion module.
If one denotes by $\Gamma$ the topological group isomorphic to $\mathbb{Z}_p^\times$, $\Gamma_n\cong\mathbb{Z}(p^n)$ cyclic of order $p^n$ and $\Lambda=\mathbb{Z}_p[[X]]=\varprojlim \mathbb{Z}_p[\Gamma_n]$, starting studying the group $\Gal(\Omega_A/\Omega_E)$, one can easily see that it is a $\mathbb{Z}(p^n)[\Gamma_n]$-module and then a $\Lambda$-module. Let $C_n=\Cl^{(p)}(K_n)$ be the $p$-primary subgroup of the ideal class group of $K_n$. Then, if $(\alpha)=\mathfrak{a}^{p^n}$ we have a homomorphism $\varphi: A_n^{1/p^n}\rightarrow C_n$, $\alpha^{1/p^n}\mapsto\mathfrak{a}$. If $u$ is a unit in $\Omega$ such that $u^{p^n}\in A_n$, then $u^{p^n}\in E_n$.
- Question 1 If $u^{p^n}\in A_n$, we have $(u^{p^n})=\mathfrak{a}^{p^n}$: how this implies $u^{p^n}\in E_n$?
Then the kernel of $\varphi$ is $E_n^{1/p^n}$ and hence we have an injective homomorphism $\psi:A_n^{1/p^n}/E_n^{1/p^n}\rightarrow C_n$, which is a $\Lambda$-homomorphism. Let us denote with $\mathscr{A}_n$ its image, i.e. we have $$ \begin{aligned} & G_{n+1} \times A_n^{1 / p^n} / E_n^{1 / p^n} \rightarrow \mu_{p^n} \\ & \quad\updownarrow\quad\quad\quad\quad\updownarrow \\ & \quad G_n \quad\times\quad \mathscr{A}_n \longrightarrow \mu_{p^n} \\ & \end{aligned}$$
- Question 2 What are the maps $G_{n+1} \times A_n^{1 / p^n} / E_n^{1 / p^n} \rightarrow \mu_{p^n}$ and $G_n \times \mathscr{A}_n \longrightarrow \mu_{p^n}$ explicitly?
Many thanks to anyone who can help me.
