Q1. Let $K$ be a local field with valuation $v$. Let us call $K'\subset K$ a nice local subfield if it is complete with respect to the induced from $K$ valuation. By local subfield I will mean a subfield $K'\subset K$ which is complete with respect to some valuation $v'$, possibly different from the one induced from $K$. Is it true that any local subfield can be embedded as a nice local subfield? In particular, I'm interested in the case when $K$ is a higher local field.


Q2. Is it true that any local subfield of a higher local field is a higher local field? Is it true in the case of nice local fields?


EDIT: As I answered below, the answer to both question is no. But of course it is true for local fields of dimension one.



Subfields of higher local fields
The answer to both questions is no, even in the case of higher local fields. 


Q1. Let us take $K = \mathbb Q\_p((T))$ and $K' = \mathbb Q\_p$. Then $K'$ is a local subfield of $K$ but there is only one embedding of $K'$ in $K$, and as a subfield of $K$, $K'$ is not a nice local subfield.


Q2. Take again $K=\mathbb Q\_p((T))$. Then $K'=\mathbb Q((T))$ is a nice local subfield but it is not a higher local field.


