"A Galois group is a fundamental group"?

Question

I read here that "a Galois group is a fundamental group". What does this mean? To every number field is there a topological space whose fundamental group is the Galois group of the polynomial?

Answer 1

That comment refers to the étale fundamental group of a scheme, which is a more subtle notion than the usual fundamental group. As stated in the comments, a thorough introduction to this point of view can be found in Szamuely's Galois Groups and Fundamental Groups (of which a preliminary version had been made generously freely available by the author).

The basic idea is that one should think of the category of finite extensions of a field $K$ as being analogous to the category of finite coverings of a topological space; the Galois group and fundamental group, respectively, come from trying to understand these categories. This analogy is closest in the case that $K$ is a one-dimensional function field over $\mathbb{C}$; in that case, it turns out that $K$ is the field of meromorphic functions on a compact Riemann surface, and that studying finite extensions of $K$ is the same thing as studying (branched) covers of this Riemann surface.

Answer 2

There is a nice survey article by Liang Xiao in here.