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I ran into the following description of Galois theory in Gelfand and Manin's Methods of Homological Algebra. But this looks like nothing like the Galois theory I know that deals with subgroups and field extensions. I wonder if there is any great self contained books that treats this topic thoroughly enough so that I can understand the following paragraphs? Thank you.

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Szamuely's book Galois Groups and Fundamental Groups has a good [edit: but apparently error-prone — see comments] exposition of Grothendieck's Galois theory, as well as its connections to geometry and topology. It's one of my favorite math books, and one of the few I've read cover to cover.

The first chapter is on the Galois theory of fields, and shows the relation between the classical formulation and Grothendieck's version. The second chapter does the same for covering spaces in topology, the third for Riemann surfaces, and the fourth for algebraic curves. The fifth chapter then explains Grothendieck's general Galois theory of schemes, which subsumes both the cases of fields and algebraic curves.

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    $\begingroup$ Your "favorite math book" contains many errors, even in fundamental definitions such as the one of a sheaf and of the étale situs. I had to prepare a seminar talk from this book about the homotopy exact sequence for the étale fundamental group. The proof in the book was a huge mess. I had to replace it with my own proof. $\endgroup$ – Martin Brandenburg Dec 11 '14 at 8:07
  • $\begingroup$ @MartinBrandenburg: I'll add that disclaimer if I recommend it in the future. (Even with some errors, I found it very helpful as a broad overview.) By the way, the definition of sheaf looks fine to me; where's the error? $\endgroup$ – Daniel Hast Dec 11 '14 at 8:16
  • $\begingroup$ Since only non-empty open subsets and non-empty intersections are considered, the sheaf condition doesn't give $F(U \sqcup V)=F(U) \times F(V)$. $\endgroup$ – Martin Brandenburg Dec 11 '14 at 8:41
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This is known as (an example of) Grothendieck's Galois theory. Lenstra's notes provide a good introduction.

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