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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
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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
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see also mathoverflow.net/questions/546/… and the Foreword in renyi.hu/~szamuely/fg.pdf –  lhf Apr 27 '11 at 20:04
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2 Answers

up vote 19 down vote accepted

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.

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.

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The second link seems to fail now. –  awllower Feb 10 at 11:42
    
Yeah, unfortunately the book is no longer available freely online from the author as far as I can tell. –  Qiaochu Yuan Feb 10 at 23:15
    
OK. Thanks still. –  awllower Feb 11 at 6:57
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There is a nice survey article by Liang Xiao in here.

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I cannot open the link. Could you provide with some key words to search? Thanks. –  awllower Feb 10 at 11:43
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OK. Thanks very much. :) –  awllower Feb 11 at 6:57
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