Completeness of field may change under the extension of field, then what justifies $K^{nr}$ is complete?

Question

Completeness of field does not change under the extension of field ?

Let $K$ be a valuation field and $v$ be one of it's valuation. Let $K$ be complete with respect to $v$.

Then, arbitrary extension field of $K$ is also complete?

I heard this is not true in some case, but I couldn't find counterexamples. If the titled question does not hold, then, the fact that $K^{nr}$($K's$ maximal unramified extension) is complete follows from
another reason?

Thank you in advance.

score 2 · Accepted Answer · answered Sep 26 '21 at 15:23

2

The maximal unramified completion of $\mathbb Q_p$ is not complete. See here for a proof.

In fact, an infinite separable extension of a complete field is never complete.

answered Sep 26 '21 at 15:23

Wojowu

26,600

Thank you！What about finite　extension case？ Completeness holds？ – Pont Sep 27 '21 at 02:28
Yes, see some references here – Wojowu Sep 27 '21 at 07:09

Completeness of field may change under the extension of field, then what justifies $K^{nr}$ is complete?

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