Galois theory for polynomials in several variables

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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?

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Take the étale fundamental group of the corresponding scheme? – Qiaochu Yuan Nov 17 2011 at 23:49
That doesn't seem like the correct generalization to me. The Galois group of $f(T)$ is the quotient of the etale fundamental group of Spec $k$ by the etale fundamental group of Spec $k[T]/(f(T))$. In particular, we would like to define it as the automorphism group of something. The Galois group of a polynomial, though, is not the automorphism group of $k[T]/f(T)$ but of $k[T_1,T_2,...,T_n]/f(T_1),etc.,etc.$. One could take the limit of the automorphism groups of $X$, some subset of $X \times X$, some subset of $X\times X\times X$, et cetera... – Will Sawin Nov 18 2011 at 2:07

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I don't understand why I see on this question a vote to close. I like this question, and vote NOT to close, and whoever has cast a vote is being very rude by not saying why. – Theo Johnson-Freyd Nov 18 2011 at 4:08

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@Theo I voted to close as not a real question. I have nothing further to add. – Felipe Voloch Nov 18 2011 at 10:20

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I should also mention that the étale fundamental group can indeed be seen as a generalization of the (absolute) Galois group for objects of dimension >0 (although this may not be the generalization you're looking for). – François Brunault Nov 18 2011 at 14:14
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2 Answers

(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.

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Thank you for the link. I'll have a look as soon as possible.  – DamienC Dec 9 2011 at 8:58

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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!)

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I don't think that increasing the number of variables means that we have to use higher category theory. – Martin Brandenburg Nov 18 2011 at 8:40
@Martin May be not, but the possibility is there. Again it is sometimes not a question of have to' but maybe 'might'. NB. In fact <i>I</i> did not mention higher category theory as the models for 2-types are quite standard simplicial things and those are classical' (due to Whitehead 1950 or Reidemeister 1930s, and Peiffer 1940s). :-) The automorphism 2-group of group is very simply the inner automorphism morphism from G to Aut(G), so is not per se higher category theory – Tim Porter Nov 18 2011 at 10:02
David Corfield kindly reminded me of this n-cat café posting:golem.ph.utexas.edu/category/2009/12/… Kim has a lot more to say on this area, but it seems that it is a very deep and hard area. In his talk at the INI that David pointed out to me, he mentioned Pursuing Stacks and the anabelian theory mentioned therein, so perhaps (more by chace than by good knowledge) I was nearer the mark than I thought! – Tim Porter Nov 20 2011 at 16:35
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