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.