Exercise 15, § 2

Let us retain the hypotheses and notation of the preceding exercise. a) Let $\alpha \in W(k)$ and $\bar{\alpha} \in W(\overline{k})$ be such that $p(\bar{\alpha}) = \alpha$ (cf. p. 45, Exercise 19). Then the element $\zeta^{p^n\bar{\alpha}}$ (p. 72, Exercise 12) of the completion $\hat{K}$ of $\overline{K}$ depends only on $\alpha$, belongs to $K$, and one has

(One may prove the equality

where $\pi = \varepsilon^{-1}(\zeta)$ (loc. cit.) and where F denotes the Frobenius automorphism of $W(k)$.) b) For $\alpha \in W(k)$, let us denote by $E**(x)$ the class in $K^*/K^{*p^n}$ of $\zeta^{p^n\bar{\alpha}}$. Then $E**(x)$ is equal to 1 if $\alpha \in \mathfrak{p}W(k)$ or if $\alpha \in p^nW(k)$. The mapping from $W_n(k)/\mathfrak{p}W_n(k)$ into $K^*/K^{*p^n}$ which associates to the class of $\alpha \in W(k)$ the element $E**(x)$ takes its values in $H_n/(K^*)^{p^n}$ and satisfies the conditions of exercise 14, b).