Galois theory for polynomials in several variables - MathOverflow

Galois theory for polynomials in several variables

2011-11-17T23:11:34Z

I feel a bit ashamed to ask the following question here.

What is (actually, is there) Galois theory for polynomials in $n$-variables for $n\geq2$?

I am preparing a large audience talk on Lie theory, and decided to start talking about symmetries and take Galois theory as a "baby" example. I know that Lie groups are somehow to differential equations what discrete groups are to algebraic equations. But I nevertheless would expect Lie (or algebraic) groups to appear naturally as higher dimensional analogs of Galois groups.

Namely, the Galois group $G_P$ of a polynomial $P(x)$ in one variable can be defined as the symmetry group of the equation $P(x)=0$ (very shortly, the subgroup of permutations of the solutions/roots that preserves any algebraic equation satisfied by them).

Then one of the great results of Galois theory is that $P(x)=0$ is solvable by radicals if and only if the group $G_P$ is solvable (meaning that its derived series reaches $\{1\}$ ).

I was wondering what is the analog of the story in higher dimension (i.e. for equations of the form $P(x_1,\dots,x_n)=0$. I would naively expect algebraic group to show up...

I googled the main key words and found this presentation: on the last slide it is written that

the task at hand is to develop a Galois theory of polynomials in two variables

This convinced me to anyway ask the question

EDIT: the first "idea" I had

I first thought about the following strategy. Consider $P(x,y)=0$ as an polynomial equation in one variable $x$ with coefficients in the field $k(y)$ of rational functions in $y$, and consider its Galois group. But then we could do the opposite...what would happen?

Answer by Tim Porter for Galois theory for polynomials in several variables

2011-11-18T08:19:14Z

This will not answer the question but is more than a comment in addition it may be very naive! (This is a hard question not a soft question!!!)

I wonder if given the Galois group <-> étale fundamental group link works for dimension 1, should there not be a link '2-Galois thingie'<->étale 2-type, and hence a link with Grothendieck's Pursuing Stacks and his letters to Breen in 1975. The sought after model might be a profinite (?) crossed module. These are able to be seen as automorphism 2-groups of groupoids, so although they are automorphism things, there is a gap to bridge before the link would work well. I have also met a similar idea when working with orbifolds, and related ideas but have not any definite reply to the particular question, rather more an addition to the question! (I hope this helps... or inspires someone to think 'outside the box'.)

There would be then a similar idea for polynomials in n-variable and models for n-types??? (This may be all rubbish but it is nice to dream sometimes!)

Answer by KristianJS for Galois theory for polynomials in several variables

2011-12-04T13:37:46Z

(This should really be a comment I think, but I'm not highly rated enough to leave one, so please bear with me)

A Galois Theoretic condition for a polynomial in two variables to be solvable by radicals is found in the following paper: http://arxiv.org/abs/math/0305226. It seems to indicate that something similar can be done for higher variables. Perhaps I'll ask Jochen next time I see him about this.