I want to prove the following result:
Proposition: Let $(K,|\;\;|)$ be a non-Archimedean valued field with residue class field $k$ such that $char(K)=char(k)=p>0$. If $K$ is locally compact, then $K$ is isomorphic to $k((x))$.
Question 1: Do you know a book or paper where I can find a proof for this proposition?
My idea of the proof use the assumption that there exists an embedding from $k$ into $K$.
Question 2: If $(K,|\;\;|)$ is a locally compact non-Archimedean valued field with residue class field $k$ such that $char(K)=char(k)=p>0$, is there a field monomorphism $\sigma:k\to K$ ?
I know the following result:
Theorem: A non-Archimedean valued field is locally compact if and only if it is Cauchy complete, its value group is discrete and its residue class field is finite.
So the proposition and the Theorem give us several hints:
- $\mathbb{F}_p$ is the prime field of $K$ and $k$.
- $k$ is isomorphic to $\mathbb{F}_q$, where $q=p^m$ and $m=[k:\mathbb{F}_p]$.
Answer to question 1: assuming the theorem in your post, you can find a proof for example in the book Corps locaux by Serre. The theorem can be found in Bourbaki.
Answer to question 2: Yes. $K$ and $k$ both contain $\mathbb{F}_p$ and the extension $k|\mathbb{F}_p$ is separable. Let $\overline{x}$ be a primitive element of this extension and let $\overline{p}\in\mathbb{F}_p[X]$ be its minimal polynomial. Lift this polynomial to a monic polynomial $p\in O[X]$ of the same degree, where $O$ is the valuation ring of $K$. In complete fields Hensels lemma holds: there exists $x\in O$ such that $p(x)=0$ and $\overline{x}$ is the residue class of $x$. Since $p$ is irreducible, the subfield $\mathbb{F}_p(x)\subset O$ is isomorphic to $k$.