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Recently, I have been studying some $p$-adic theory from Gouvêa's p-adic Numbers, an Introduction, and I know that $\mathbb{Q}_p$ is a completion of the rationals with the $p$-adic absolute value. However, while skimming, I realized upon the fact that the algebraic closure of the p-adic numbers $\overline{\mathbb{Q}}_p$ is not complete, and that to get a true analogue of the complex numbers, we would have to take the completion of $\overline{\mathbb{Q}}_p$.

I then had a thought: are there any (possibly pathological) cases of fields where taking an algebraic closure results in it not being complete, and trying to complete it results in it not being algebraically closed, etc. I'm not even sure if this is that great of a question, but any insights would be appreciated!

abiteofdata
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    This might be an interesting question, but it needs to be formalized. In particular, while being alg. closed is a property just of a field, and every field has a kind-of-unique algebraic closure, the concept of completion depends on the choice of (a little more than) a topology. Further, to decide whether "the" algebraic closure is complete, we have to somehow extend the topology of the prior field to such an algebraic closure in a non-arbitrary way. One might doubt if that is always possible, to begin with. – Torsten Schoeneberg Oct 17 '22 at 03:55
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    With these thoughts, in my mind the question becomes something like: Is there a topological field $K$ such that for any algebraically closed extension $E\vert K$, there is no topology on $E$ satisfying ??? (here, formalize the notion that the topology on $E$ somehow extends the one on $K$), and such that $E$ is complete? – Torsten Schoeneberg Oct 17 '22 at 04:00
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    Have a look at Kürschak's theorem, mentioned here: https://math.stackexchange.com/questions/123925/is-the-algebraic-closure-of-a-p-adic-field-complete/123962#123962 – Brauer Suzuki Oct 17 '22 at 04:35

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