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Using the theorem of Puiseux, one concludes that the algebraic closure of $\mathbb C(X)$ is the set of algebraic elements (over $\mathbb C(X)$) of the algebraic closure of $\mathbb C((X))$, which is $\cup_n\mathbb C((X^{1/n}))$. I have two questions:

  • Is there any direct description of this field?
  • What is the Galois group $\mathrm{Gal}(\overline{\mathbb C(X)}/\mathbb C(X))$?

Thanks.

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    $\begingroup$ What do you mean by «direct» description? $\bigcup_n\mathbb C((X^{1/n}))$ seems pretty direct to me… $\endgroup$ May 12, 2015 at 7:43
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    $\begingroup$ @AmitH: I think you meant the absolute Galois group of $\mathbb{C}((X))$. Every finite group appears as Galois group over $\mathbb{C}(X)$, by the Riemann existence theorem. $\endgroup$ May 12, 2015 at 9:05
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    $\begingroup$ The Galois group of $\overline{\mathbb{C}(X)}$ over $\mathbb{C}(X)$ is the free profinite group over the set $\mathbb{C}$. See for instance "Algèbre et théories galoisiennes" by R. and A. Douady, no. 6.4. $\endgroup$
    – abx
    May 12, 2015 at 9:06
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    $\begingroup$ @Loïc Teyssier, $\cup_n\mathbb C((X^{1/n}))$ is not the algebraic closure of $\mathbb C(X)$ $\endgroup$
    – Z.A.Z.Z
    May 12, 2015 at 10:38
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    $\begingroup$ @AllyMath, sorry, I read the sentence too fast. $\endgroup$ May 12, 2015 at 17:06

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