Locally compact valued field is complete?

Question

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?

score 8 · Answer 1 · answered Sep 29 '17 at 12:42

8

You are right: a locally compact metric space isn't necessarily complete. But every metrizable locally compact group is complete.

answered Sep 29 '17 at 12:42

José Carlos Santos

440,053

Is there a reference for this please? – PunkZebra Sep 13 '23 at 15:26
@PunkZebra Check theorem 4.5 from this article. – José Carlos Santos Sep 13 '23 at 15:33

score 4 · Answer 2 · answered Sep 29 '17 at 13:03

4

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.)

answered Sep 29 '17 at 13:03

Andres Mejia

21,467

Locally compact valued field is complete?

2 Answers2

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