Extensions of the field $k((x))$ of Laurent series

Question

Let $k((x))$ be the field of Laurent series over some field $k$. Is it always true that a finite extension $L$ of $k((x))$ is again a field of Laurent series $k'((y))$ over some finite extension $k'$ of $k$?

I guess this does hold for instance if $k$ is a finite field or if $k$ is of characteristic zero but I don't quite understand the degree $p$ extensions of $k((x))$ in characteristic $p$. Is it still true for arbitrary $k$?

I would be thankful for help and/or a reference.

Answer 1

I found a reference by Patrick Allen in which one finds for a field $K$ that is complete with respect to a non-trivial valuation $v$ the following:

Theorem 5.16. If $v$ is discrete and $K$ has equal characteristic, then $K \cong k((T ))$ as valued fields.