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Let $(K, |\cdot |)$ be a (discrete) valuation field, and $(\widehat{K}, |\cdot|)$ be its completion. Then we can think about the algebraic (or separable) closure of $K$ in $\widehat{K}$, which is called Henselization of $K$.

I want to know how to describe the Henselization of a given field. For example. Let's assume that we have a fixed (odd?) prime $p$ and $K = \mathbb{Q}$ with a $p$-adic norm $|\cdot |_{p}$. What is the Henselization, i.e. algebraic closure of $\mathbb{Q}$ in $\mathbb{Q}_{p}$? This is strictly smaller than $\mathbb{Q}_{p}$ since not every element in $\mathbb{Q}_p$ is algebraic over $\mathbb{Q}$. Also, we can find some nontrivial examples which are in the Henselization. For example, if $p = 5$ then $\sqrt{11}$ is in $\mathbb{Q}_{5}$ and so in the Henselization. Is there an explicit way to describe elements in that field? Do we have some set-theoretical issue here?

Seewoo Lee
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  • Why would there be any set-theoretic issues? – Qiaochu Yuan Apr 08 '19 at 21:55
  • @QiaochuYuan Actually it is not an issue, but I just want to talk about something like Hamel basis, which we know the existence (assuming AC) but there's no way to construct it explicitly. – Seewoo Lee Apr 08 '19 at 23:23
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    I don't really understand what would count as a good description for you, here. Are you satisfied with the description of $\overline{\mathbb{Q}}$ as the algebraic closure of $\mathbb{Q}$, despite e.g. not being able to write down a Hamel basis of it? Would you be satisfied with knowing which polynomials over $\mathbb{Q}$ have roots in $\mathbb{Q}_p$? That shouldn't be hard to settle using Hensel's lemma (which I think is why this is called the Henselization). – Qiaochu Yuan Apr 09 '19 at 03:57
  • Just a few days ago there was this question with the exact same title: https://math.stackexchange.com/q/3172907/96384. While I cannot really advertise my near-trivial answer there, the comments contain a link to this question and its answers, and linked from there is this helpful MathOverflow post: https://mathoverflow.net/q/17032/27465. – Torsten Schoeneberg Apr 09 '19 at 04:32
  • @QiaochuYuan Now I understand that I asked a stupid question. – Seewoo Lee Apr 10 '19 at 00:32
  • @TorstenSchoeneberg Thank you for a good answer! – Seewoo Lee Apr 10 '19 at 00:32

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