# Complete fields with algebraically closed residue field

I am looking for a reference where the following result is proven:

Let $k$ be an algebraically closed field. If $K$ is a complete and discretely valued field with residue field $k$. Then $K$ is one of the following:

1) The field of Laurent series in $k$.

2) A finite and totally ramified extension of a field of Witt vectors with components in $k$.

On another note, Lang later calls any such field from the list as the complete unramified field with residue field $k$. Can anyone explain me what does he mean by an unramified field (I am only familiar with what an unramified extension is) ? For example, one page later, Lang claims that if we have two such complete unramified fields $K_1\subset K_2$ with respective algebraically closed residue fields $k_1\subset k_2$, then for an intermediate finite extension $K_1\subset E$, with $E/K_1$ totally ramified, one must have $E=K_1$. I have really tried to find answers for some time on my own, but with no luck.

• In mixed characteristic, a discretely valued field $K$ is called unramified if $p$ is a uniformiser in $\mathcal O_K$. I'm not sure if there is a similar notion in pure characteristic, because there is no initial object in this category. – R. van Dobben de Bruyn Apr 4 '17 at 23:03
• Some of this material is covered in Tag 09E3, where this notion is called weakly unramified, and the extension is unramified if moreover the residue extension is separable. This does not agree with the general notion of unramified ring morphisms, because those are usually assumed of finite type, cf. Tag 00US. In other sources, you may find different terminology. – R. van Dobben de Bruyn Apr 4 '17 at 23:10

## 1 Answer

Serre, Local fields, Chapter II: II.4 for the first case and II.5 for the second case.