The question adresses the answer in this thread: Algebraic closure of $k((t))$
In the answer reuns used a theory relating classical Galois theory with Galois theory of ramified coverings. I'm an absolute newbie in this area, so sorry if it's to simple. What I understand:
In this case we consider a finite Galois extension $L/\Bbb{C}(z)$. To field $L$ the theory says that one can associate a unique connected Riemann surface $Y_L$ and a map $f_L: Y_L \to \Bbb{PC^1}$ which is a ramified cover over $\Bbb{PC^1}$ of degree $n=[L:\Bbb{C}(z)]$.
Ramified means that that there exist a finite subset $S \subset \Bbb{PC^1}$ (= the "branch points") such that the restricted map $Y_L \backslash f_L^{-1}(S) \to \Bbb{PC^1} \backslash S$ is a $n$-cover known from basic topology.
We recover $L$ as meromorphic functions on $Y_L$ (and $\Bbb{C}(z)$ " $\Bbb{PC^1}$).
The deepth of the theory gives a beautiful identification for Galois group $$Gal(L/\Bbb{C}(z))= \pi_1(\Bbb{PC^1} \backslash S,z_0)/Q$$
where $\pi_1(\Bbb{PC^1} \backslash S,z_0)$ is the fundamental group and the subgroup $Q \subset \pi_1(\Bbb{PC^1}\backslash S,z_0)$ corresponds via classical covering theory to the cover $p:U_{\Bbb{PC^1}\backslash S} \to Y_L \backslash f_L^{-1}(S)$. Here $U_{\Bbb{PC^1}\backslash S}$ is the universal cover of $\Bbb{PC^1}\backslash S$. That's the background.
Two questions:
reuns wrote:
For $L/\Bbb{C}(z)$ a finite Galois extension then its elements are locally meromorphic on $\Bbb{C}$ minus a few branch points (the zeros of the discriminant of the minimal polynomials) and $Gal(L/\Bbb{C}(z))$ consists of the analytic continuations along closed loops enclosing some of those branch points.
Proof: with $\gamma_1(a),\ldots,\gamma_m(a)$ the analytic continuations of $a$ then the coefficients of $h(X)=\prod_{l=1}^m(X-\gamma_m(a))$ stay the same under analytic continuation, thus they are meromorphic on the Riemann sphere, ie. they are in $\Bbb{C}(z)$ so $h(X)$ is the minimal polynomial of $a$.
QUESTION #1: I learned this theory from Szamuely's "Galois Groups and Fundamental Groups". In the construction there wasn't explicitly explaned that the ramified points of $f$ correspond exacly to the zeros of the discriminant of minimal polynomial $F$ of the generator $g \in L$ (in case $L= \Bbb{C}(z)(g)$, otherwise consider minpolys of all generators).
Does anybody know literature explaning this construction of $Y_L$ from $L$ where is explicitly explaned why the branch points are exactly the zeros of the discriminant of minimal polynomials.
QUESTION #2:
How a $l \in \pi_1(\Bbb{PC^1}\backslash S,z_0)$ represents an element of the Galois group $Gal(L/\Bbb{C}(z))$. Szmuely's book proof's the isomorphism $\pi_1(\Bbb{PC^1}\backslash S,z_0)/Q= Gal(L/\Bbb{C}(z))$ but unfortunately the proof not gives a good geometric inside look what is going on there explicitely. In other words I "understand" the proof but have no visual intuition on this identification.
What we need is that such $l$ act as automorphism on the meromorphic function from $L$ and fixes $\Bbb{C}(z)$.
Reuns explaned this identification as " analytic continuations along closed loops enclosing some of those branch points."
I would like to understand what he means here. ie wlog say our $l$ is a class of such closed loop around a pranch point. What/ which object is here "analytically continued along $l$"? How this construction works. The object of our interest is the meromorphic function $m \in L$ of $Y_L$.
How it can be "continued"? Isn't it already determined at a dense open subset of $Y_L$ exept at it's poles. Thus why and how it is continued along $l$? I not understand it.