Let $K$ be a field and $L$ a finite extension. Let furthermore $L$ be complete regarding a (non-Archimedian) norm $|\cdot|$ (with non-trivial restriction $|\cdot|_{|K}$). Is then $K$ also complete? In formulas:
$[L:K]<\infty$ and $L$ complete $\Rightarrow K$ is complete?
The converse is true, i.e. if $K$ is complete, then $L$ is complete, but I don't know if the other way round is true too. So does anyone know about this statement and maybe even knows a proof or counterexample?
If the norm is Archimedian the statement is true, since we can apply Ostrowski to get $L \cong \mathbb R$ or $\mathbb C$ and then get by Artin-Schreier that $K\cong \mathbb R$ or $\mathbb C$.
In general it seems to be true too, since we can see $L$ as finite vector space over $K$ and hence have $K$ as 1-dimensional subspace in a complete vector space, so maybe we can solve it topologically from here?
PS: Is this better suited for here or overflow?