Exercise 7, § 5

Let K be a field and v a discrete valuation on K such that K is complete with $\mathscr{T}_v$: let $\mathbf{A}$ and $\mathfrak{p}$ denote the ring and ideal of $v$ and $U = \mathbf{A} - \mathfrak{p}$ the set of invertible elements of $\mathbf{A}$. Let $u$ be an isomorphism of $K$ onto a subfield of $K$.

(a) Show that $u(\mathfrak{p}) \subset \mathbf{A}$ and $u(U) \subset U$. (To prove the first point, observe that for all $z \in \mathfrak{p}$ the equation $x^n = 1 + z$ admits a solution in $\mathbf{A}$ for all $n > 0$ prime to the characteristic of the residue field of $v$, using Hensel's Lemma; if $v(u(z)) < 0$, deduce that the integer $v(u(z))$ would be divisible by every integer $n > 0$ prime to the characteristic of the residue field of $v$. Then deduce the second assertion from the first.)

(b) Deduce from (a) that, either $u(K^*) \subset U$, or $u$ is continuous (consider the image under $u$ of a uniformizer of $v$).

(c) Give an example of an algebraically closed field $\Omega$ such that, taking $K = \Omega((X))$, $K$ is isomorphic to a subfield of $K$ contained in $U \cup \{0\}$ (cf. Algebra, Chapter V, § 5, Exercise 13).