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A while ago when I was learning Galois Theory for the first time, I asked my professor if there was an analogue of field extensions/Galois Theory for polynomials of $n \geq 2$ variables. The best I got was a "yes, but it's tough".

I know that the typical approach doesn't really work, even for $n=2$. As if $f(x,y) \in k[x,y]$ is an irreducible polynomial, then the quotient corresponds to the coordinate ring of the variety defined by the zero set of $f$, and is not a field.

A quick google on this returned a mathoverflow thread:

https://mathoverflow.net/questions/81209/galois-theory-for-polynomials-in-several-variables

but the discussion uses far more algebraic geometry than I know, and from what I could gather the discussion there didn't answer the question fully. Is there an analogue of field extensions and Galois Theory for multivariate polynomials? If anyone has a first-year grad student level explanation, I'd very much appreciate it.

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    The example polynomial you have there does not result in a field (it is not even a prime ideal). The thing that happens is that you need more than one polynomial. – 2017-02-09
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    You are correct. Edited. – 2017-02-09
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    Most ring of integers are not [monogenic](https://en.wikipedia.org/wiki/Monogenic_field). If generated by two elements then $\mathcal{O}_K = \mathbb{Z}[\alpha,\beta] \simeq \mathbb{Z}[x,y]/(f(x),g(x,y))$ where $f(x)$ is the minimal polynomial of $\alpha$ over $\mathbb{Q}$ and $g(\alpha,y)$ is the minimal polynomial of $\beta$ over $\mathbb{Q}(\alpha)$ – 2017-02-09
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    Well, there are many generalizations of Galois theory to higher dimensions, and I am not sure I understand precisely the generalization you are looking for (nor I am not sure that it exists !). In the comments of the linked mathoverflow thread, there is a discussion about the étale fundamental group and the Galois theory of étale covers. This is indeed a generalization. It is of higher dimension because the base scheme is not the spectrum of a field (so not of dimension 0) but can be arbitrary (so for example the spectrum of $k[x,y]/(f)$). Are you looking for something in this direction ? – 2017-02-09
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    I guess I'm not exactly sure what I'm asking for. When I originally asked the question it was in the context of extensions of fields by roots of polynomials, and I had wondered if such a thing were possible for polynomials of two variables. So I would say something along the lines of solvability for multivariate polynomials. I realize this may be a naive inquiry, and possibly ill-defined, but any information you may have is welcome. – 2017-02-09

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