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.