Exercise 14, § 2

Let $A$ be a complete discrete valuation ring of characteristic zero, whose residue field $k$ is perfect of characteristic $p$. Let $K$ be the field of fractions of $A$. Let $n$ be an integer $\geq 1$ and let $\zeta$ be a primitive $p^n$-th root of unity in $K$. Let $\overline{K}$ be an algebraic closure of $K$, endowed with the unique valuation $v$ extending that of $K$.

a) Let $C$ be the Cohen subring of $A$, $T$ its field of fractions; it is a subfield of $K$. The canonical isomorphism of $W(k)$ onto $C$ induces an isomorphism of $W(k) \otimes_{\mathbf{Z}} \mathbf{Q}$ onto $T$. For every finite degree extension $l$ of $k$, the field $W(l) \otimes_{\mathbf{Z}} \mathbf{Q}$ is a finite degree extension of $W(k) \otimes_{\mathbf{Z}} \mathbf{Q}$, galois if $l$ is so. If $l$ is galois, the unique extension $T_l$ of $T$ in $\overline{K}$ isomorphic to $W(l) \otimes_{\mathbf{Z}} \mathbf{Q}$ has a residue field isomorphic to $l$. In particular the field $T^\infty$, union of the fields $T_l$ when $l$ runs through the galois finite degree extensions of $k$ (in a fixed algebraic closure of $k$), has as residue field an algebraic closure $\overline{k}$ of $k$. The same is true of $K$. For each extension $l$ of $k$ in $\overline{k}$, we shall denote by $T_l$ the unique extension of $T$ in $T^\infty$ whose residue field is the subfield $l$ of $k$, and by $K_l$ the compositum of $K$ and $T_l$. Then, if $l$ is galois over $k$, $K_l$ is galois over $K$ and its Galois group is canonically identified with the Galois group of $l$ over $k$.

b) Let $k_n$ be the largest abelian extension of $k$ in $\overline{k}$ whose galois group is annihilated by $p^n$. By exerc. 19, p. 45, we have a canonical isomorphism

By A, V, p. 85, th. 4, we have a canonical isomorphism

where $H_n = (K_{k_n}^*)^{p^n} \cap K^*$.

Prove that there exists a unique mapping

such that the following diagram is commutative:

where the right vertical arrow denotes the mapping which associates with the homomorphism $\varphi$ of $H_n/K^{*p^n}$ into $\mu_{p^n}(K)$ the homomorphism $\psi$ of $W_n(k)/\mathfrak{p}W_n(k)$ into $\mathbf{Z}/p^n\mathbf{Z}$ defined by $\zeta^{\psi(\alpha)} = \varphi(E^*(\alpha))$ for $\alpha \in W_n(k)/\mathfrak{p}W_n(k)$. The mapping $E^*$ is an isomorphism.