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:
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> 
> 1) The field of Laurent series in $k$.
> 
> 
> 2) A finite and totally ramified extension of a field of Witt vectors with components in $k$.
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> 
> 


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.



Complete fields with algebraically closed residue field
Serre, Local fields, Chapter II: II.4 for the first case and II.5 for the second case.


