Exercise 5, § 9

(a) Suppose that K is a commutative field, $v$ a valuation of height 1 on K such that K is complete with respect to $v$ and $\mathbf{A}$ the ring of $v$; suppose further that the residue field $k$ of $v$ is perfect and of characteristic $p > 0$. For every element $\xi \in k^*$ and every integer $n$, let $x_n$ be an element of the class $\xi^{p^{-n}}$ in $\mathbf{A}$; show that the sequence $(x_n^{p^n})$ is a Cauchy sequence in $\mathbf{A}$ whose limit is independent of the choice of the $x_n$ in the classes $\xi^{p^{-n}}$. If this limit is denoted by $\phi(\xi)$, show that $\phi$ is the unique isomorphism $u$ of the multiplicative group $k^*$ to the multiplicative group $K^*$ such that, for all $\xi \in k^*$, $u(\xi)$ is an element of $\mathbf{A}$ belonging to the class $\xi$.

(b) If K is also of characteristic $p$, show that $\phi$, extended to $k$ by $\phi(0) = 0$, is an isomorphism of the field $k$ onto a subfield of K. Deduce a new proof of Theorem 1 (iii) of no. 3.

(c) Suppose that $k$ is finite. Show that, if $r$ is prime top, the group $(K^*)^r$ of $r$-th powers of elements of $K^*$ is of finite index in $K^*$ (use Hensel's Lemma). Show that, if further $v$ is discrete and K of characteristic 0, the same result is valid without restriction on $v$ (observe that every element of $1 + p^2\mathbf{A}$ is a $p$-th power).