Exercise 20, § 1

Retain the hypotheses and notation of the preceding exercise.

a) For each extension L of K in $\Omega$, consider the $\mathbf{Z}_p$-module

and, for each integer $n \geqslant 0$, its submodule

Prove that $\mathscr{W}(L)$ can be identified with the inductive limit of its submodules $\mathscr{W}_n(L)$ with respect to the inclusion mappings

b) For every integer $n \geqslant 0$, let $\psi_n$ be the mapping of $\mathscr{W}_n(L)$ into $W_n(L)/\mathfrak{p} W_n(L)$ which associates to $x \otimes p^{-n}$ the class of $x$. Prove that $\psi_n$ is an isomorphism and that one has

where $V_n$ is the mapping of $W_n(L)/\mathfrak{p} W_n(L)$ into $W_{n+1}(L)/\mathfrak{p} W_{n+1}(L)$ induced by the shift V. Deduce that $\mathscr{W}(L)$ can be identified with the inductive limit of the groups $W_n(L)/\mathfrak{p} W_n(L)$ with respect to the mappings $V_n$.

c) Let $a \in \mathscr{W}(K)$. If $n$ is an integer such that $a$ belongs to $\mathscr{W}_n(K)$, the extension $K(p^{-1}(\psi_n(a)))$ was constructed in Exercise 19, c). This extension does not depend on the choice of the integer $n$; it is denoted by $K(p^{-1}(a))$. If $A$ is a subset of $\mathscr{W}(K)$, we denote by $K(p^{-1}(A))$ the extension generated by the fields $K(p^{-1}(a))$ for $a$ running through $A$. Prove that $K(p^{-1}(A))$ is an abelian extension of $K$. Prove, in this situation, assertions analogous to assertions (i) to (iv) of Exercise 19, c).