# “A Galois group is a fundamental group”?

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?

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I am not sure, any group is a fundamental group of a CW-complex, so "being a fundamental group" is not an interesting property... – Thomas Rot Apr 27 '11 at 19:56
Perhaps the comment was referring to the algebraic geometry version of the fundamental group? Check wikipedia for "Etale fundamental group". – MartianInvader Apr 27 '11 at 20:00
see also mathoverflow.net/questions/546/… and the Foreword in renyi.hu/~szamuely/fg.pdf – lhf Apr 27 '11 at 20:04

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.