Exercise 21, § 1

Let us retain the hypotheses and notations of the two preceding exercises. a) Multiplication by $p$ in $W(K)/p\mathscr{W}(K)$ is injective. b) Let $a$ be an element of $W(\Omega)$ such that $a \notin W(K)$ and $pa \in W(K)$. Denote by $K(a)$ the subfield of $\Omega$ generated over $K$ by the components $(a_n)_{n \in \mathbf{N}}$ of $a$. Prove that $K(a)$ is an abelian extension of $K$, whose galois group is isomorphic to the topological group $\mathbf{Z}_p$. c) If $A$ is a subset of $W(K)$, denote by $K(p^{-1}(A))$ the subextension of $\Omega$ generated by the fields $K(a)$ for all elements $a$ of $W(\Omega)$ such that $pa \in A$. Prove in this situation assertions analogous to assertions (i) to (iv) of exerc. 19, c). d) Let $\Gamma$ be the automorphism group of $\Omega$ over $K$. Let $n$ be an integer $\geqslant 1$ and let $\varphi_n$ be a continuous homomorphism of $\Gamma$ into $\mathbf{Z}/p^n\mathbf{Z}$. Prove that there exists a continuous homomorphism $\varphi$ of $\Gamma$ into $\mathbf{Z}_p$ which induces $\varphi_n$ by passing to the quotient. Denoting by $\mathrm{Hom}_c(\Gamma, \Gamma')$ the group of continuous homomorphisms of $\Gamma$ into $\Gamma'$, $\Gamma'$ being a topological group, prove that the canonical mapping

is an isomorphism.