Let $L$ and $K$ be two complete discrete valuation fields of equal characteristic $0$. Assume that an embedding $K\subset L$ is fixed and let $F$ be the algebraic closure of $K$ inside $L$. Can you provide a proof or at least an hint for the following proposition?
Proposition: $F$ is a finite extension of $K$ and $L\cong F((t))$.
In particular this means that $F$ is actually the residue field of $L$. why is this true?
Edit: On $L$ we have an additional hypothesis. We know that its residue field $\overline L$ is again a complete discrete valuation field (of characteristic $0$). In other words $L$ has "dimension $2$''. For example $L=\mathbb Q_p((t))$.
Thank you in advance