Exercise 19, § 1

Let K be a field of characteristic $p$, P its prime subfield, $\Omega$ an algebraic closure of K. We denote by $\wp$ the endomorphism $x \mapsto Fx - x$ of the group $W(\Omega)$, and also, for every integer $n \geqslant 1$, the endomorphism of $W_n(\Omega)$ which it induces by passing to quotients.

a) Let us fix an integer $n \geqslant 1$. The endomorphism $\wp$ of $W(\Omega)$ (resp. $W_n(\Omega)$) leaves $W(K)$ (resp. $W_n(K)$) stable and its kernel is $W(P)$ (resp. $W_n(P)$).

In the sequel to this exercise, $W(P)$ and $\mathbf{Z}_p$, $W_n(P)$ and $\mathbf{Z}/p^n\mathbf{Z}$ will be identified (§ 1, No. 7, Example 3).

b) Let $a \in W_n(K)$. Denoting by V the shift homomorphism $W_n(K) \to W_{n+1}(K)$, prove that one has $a \in \mathfrak{p} W_n(K)$ if and only if one has $Va \in \mathfrak{p} W_{n+1}(K)$. If K is separably closed, one has $\mathfrak{p} W_n(K) = W_n(K)$.

c) Let $a \in W_n(\Omega)$ be such that $\mathfrak{p} a \in W_n(K)$. We denote by $K(a)$ the subfield of $\Omega$ generated by K and the components $a_0, ..., a_{n-1}$ of $a$. If A is a subset of $W_n(K)$, we denote by $K(\mathfrak{p}^{-1}(A))$ the subextension of $\Omega$ generated by the fields $K(a)$, for all elements $a$ of $W_n(\Omega)$ satisfying $\mathfrak{p} a \in A$.

Reasoning as in A, V, p. 87 and using the preceding exercise, prove the following assertions:

(i) Let L be a Galois extension of K in $\Omega$. There exists a unique mapping $(\sigma, a) \mapsto [\sigma, a]$ of $\mathrm{Gal}(L/K) \times (\mathfrak{p} W_n(L) \cap W_n(K))/\mathfrak{p} W_n(K)$ into $\mathbf{Z}/p^n\mathbf{Z}$ such that for every $\sigma \in \mathrm{Gal}(L/K)$ and every $x \in W_n(L)$ such that $\mathfrak{p}(x) \in W_n(K)$, one has, denoting by $\overline{\mathfrak{p}(x)}$ the class of $\mathfrak{p}(x)$ modulo $\mathfrak{p} W_n(K)$

If $\sigma, \sigma' \in \mathrm{Gal}(L/K)$ and $a, a' \in (\mathfrak{p} W_n(L) \cap W_n(K))/\mathfrak{p} W_n(K)$, one has

and

(ii) Let us denote, for every Galois extension L of K in $\Omega$,

and

the homomorphisms deduced from the mapping $(\sigma, a) \mapsto [\sigma, a]$ constructed in (i). For every Galois extension L of K in $\Omega$, the homomorphism $a_L$ is injective, and its image is the group of continuous homomorphisms of the topological group $\mathrm{Gal}(L/K)$ into the discrete group $\mathbf{Z}/p^n\mathbf{Z}$.

(iii) The mapping $A \mapsto K(\mathfrak{p}^{-1}(A))$ is a bijection of the set of subgroups of $W_n(K)$ containing $\mathfrak{p} W_n(K)$ onto the set of extensions of K in $\Omega$, abelian over K and of exponent dividing $p^n$. The inverse mapping is $L \mapsto \mathfrak{p} W_n(L) \cap W_n(K)$.

(iv) For every subgroup A of $W_n(K)$ containing $\mathfrak{p} W_n(K)$, the homomorphism

is bijective. When one endows $\mathrm{Hom}(A/\mathfrak{p} W_n(K), \mathbf{Z}/p^n\mathbf{Z})$ with the topology of simple convergence, it is a homeomorphism.