TITLE: Locally compact valued field is complete?
QUESTION [5 upvotes]: In J. P. Serre's book Local Fields, in the proof of one proposition it says:
If $K$ is locally compact, it is complete.
How did he deduce that? As far as I know, there is no correlation between locally compactness and completion. So I assume he used the fact that $K$ is a valued field, but how?

REPLY [4 votes]: Hint: Let $K$ be a locally compact topological field and $\overline{K}$ its completion. Then $K$ is a subfield of $\overline K$, and hence closed (so in particular, complete.)