That is a precise copy of the mse query https://math.stackexchange.com/questions/3673802/galois-theory-of-ramified-coverings-classical-galois-theory?noredirect=1#remark7550087_3673802 the place I sadly not received the reply I used to be on the lookout for.

The query adresses the reply on this thread: Algebraic closure of $ok((t))$

Within the reply reuns used a principle relating classical Galois principle with Galois principle of ramified coverings. I am an absolute beginner on this space, so sorry if it is to easy. What I perceive:

On this case we take into account a finite Galois extension $L/Bbb{C}(z)$. To subject $L$ the speculation says that one can affiliate a novel linked Riemann floor $Y_L$ and a map $f_L: Y_L to Bbb{PC^1}$ which is a ramified cowl over $Bbb{PC^1}$ of diploma $n=[L:Bbb{C}(z)]$.

Ramified implies that that there exist a finite subset $S subset Bbb{PC^1}$ (= the “department factors”) such that the restricted map $Y_L backslash f_L^{-1}(S) to Bbb{PC^1} backslash S$ is a $n$-cover recognized from primary topology.

We get well $L$ as meromorphic capabilities on $Y_L$ (and $Bbb{C}(z)$ ” $Bbb{PC^1}$).

The deepth of the speculation offers an exquisite identification for Galois group $$Gal(L/Bbb{C}(z))= pi_1(Bbb{PC^1} backslash S,z_0)/Q$$

the place $pi_1(Bbb{PC^1} backslash S,z_0)$ is the elemental group and the subgroup $Q subset pi_1(Bbb{PC^1}backslash S,z_0)$ corresponds through classical overlaying principle to the quilt $p:U_{Bbb{PC^1}backslash S} to Y_L backslash f_L^{-1}(S)$. Right here $U_{Bbb{PC^1}backslash S}$ is the common cowl of $Bbb{PC^1}backslash S$. That is the background.

Two questions:

reuns wrote:

For $L/Bbb{C}(z)$ a finite Galois extension then its parts are domestically meromorphic on $Bbb{C}$ minus just a few department factors (the zeros of the discriminant of the minimal polynomials) and $Gal(L/Bbb{C}(z))$ consists of the analytic continuations alongside closed loops enclosing a few of these department factors.

Proof: with $gamma_1(a),ldots,gamma_m(a)$ the analytic continuations of $a$ then the coefficients of $h(X)=prod_{l=1}^m(X-gamma_m(a))$ keep the identical underneath analytic continuation, thus they’re meromorphic on the Riemann sphere, ie. they’re in $Bbb{C}(z)$ so $h(X)$ is the minimal polynomial of $a$.

**QUESTION #1:** I realized this principle from Szamuely’s “Galois Teams and Basic Teams”. Within the building there wasn’t explicitly explaned why the department factors of $f_L$ are the widespread zeros of the discriminants of minimal polynomials $F_i(z,T)$ of the mills $g_i in L$ (in case $L= Bbb{C}(z)(g)$ of $L$.

Does anyone know literature explaning this building of $Y_L$ from $L$ in that method there’s *explicitly* identified why the department factors are precisely the widespread zeros of the discriminants $Disc(F_i(z,T))$ of minimal polynomials.

**QUESTION #2:**

How a $l in pi_1(Bbb{PC^1}backslash S,z_0)$ represents

a component of the Galois group $Gal(L/Bbb{C}(z))$ explicitely?

Szmuely’s e book proves the isomorphism

$pi_1(Bbb{PC^1}backslash S,z_0)/Q= Gal(L/Bbb{C}(z))$

utilizing class equivalences and sadly this proof not offers a superb geometric

inside look what’s going on there. In different phrases I “perceive”

the proof step-by-step however haven’t any visible instinct on this identification.

What we’d like is that such $l$ act as automorphism on

the meromorphic operate from $L$ and fixes $Bbb{C}(z)$.

Reuns explaned this identification as

” analytic continuations alongside closed loops enclosing a few of

these department factors.”

Sadly I am unable to think about what he means by this analytic continuation and I wish to perceive what he means right here. ie wlog say our

$l$ is a category of such closed loop round a pranch level.

What/ which object is right here “analytically continued alongside $l$“?

How this building works. The thing of our curiosity

is a meromorphic operate $m in L$ of $Y_L$.

How it may be “continued”? Is not it already decided

at a dense open subset of $Y_L$ exept at it is poles.

Thus why and the way it’s continued alongside $l$? I not

perceive it.