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Let $K$ be a field, and $x$ an element transcendental over $K$.

We know that $K(x)$ is not algebraically closed, in fact it is enough to consider the polynomial $t^2 - x$ (Algebraic closure of $K(X)$).

Is there a way to explicitly describe $\overline{K(x)}$?

Jsevillamol
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Kalhac
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    As soon as you start looking at $K(x)$, you’re dealing with geometry. You are asking, among many other things, what all the algebraic curves over $K$ are. – Lubin Jan 17 '18 at 19:11
  • On the other hand, the field of Puiseux series is the algebraic closure of $k((x))$ if $k$ is an algebraically closed field of characteristic 0. – Watson Jul 14 '18 at 11:24
  • For $K=\Bbb C$, see https://math.stackexchange.com/questions/59826 – Watson Aug 15 '18 at 21:11

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