# An advanced exposition of Galois theory

My knowledge of Galois theory is woefully inadequate. Thus, I'd be interested in an exposition that assumes little knowledge of Galois theory, but is advanced in other respects. For instance, it would be nice if it were to include remarks like the following:

A finite field extension $K / k$ is separable iff the geometric fiber of Spec k -> Spec K is a finite union of reduced points.

[I was never able to remember what "separable" meant until I saw this equivalence while studying unramified morphisms. The proof is by the Chinese Remainder Theorem. Also note: this definition is incomplete, in the sense that it does not specify when a non-finite extension is separable.]

Is there any such exposition?

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"Galois theories" by Francis Borceux and George Janelidze is quite nice. (books.google.com/…) –  user717 Jul 13 '10 at 7:55
Read the discussion of "Galois categories" in SGA1. –  BCnrd Jul 13 '10 at 15:39

## 5 Answers

I recommend H. W. Lenstra's Galois theory for schemes.

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That book looks interesting, but explicitly assumes knowledge of Galois theory for finite extensions. –  Charles Staats Jul 13 '10 at 3:15

Szamuely's Galois Groups and Fundamental Groups might be what you're looking for. In particular, the beginning of Chapter 2 (where the discussion switches from field theory to fundamental groups) alludes to a statement like the one you give:

In the last section we saw that when studying extensions of some ﬁeld it is plausible to conceive the base ﬁeld as a point and a ﬁnite separable extension (or, more generally, a ﬁnite etale algebra) as a ﬁnite discrete set of points mapping to this base point. Galois theory then equips the situation with a continuous action of the absolute Galois group which leaves the base point ﬁxed. It is natural to try to extend this situation by taking as a base not just a point but a more general topological space. The role of ﬁeld extensions would then be played by certain con- tinuous surjections, called covers, whose ﬁbres are ﬁnite (or, even more generally, arbitrary discrete) spaces. We shall see in this chapter that under some restrictions on the base space one can develop a topological analogue of the Galois theory of ﬁelds, the part of the absolute Galois group being taken by the fundamental group of the base space.

Edit: I notice that this book was discussed in another MO question here: Galois Groups vs. Fundamental Groups .

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Can anyone recommend (or anti-recommend) this book based on experience? –  Charles Staats Jul 13 '10 at 0:37
I partially can. Szamuely discusses some aspects of Dedekind schemes which were quite enlightening for me. You can take a look at this chapter here: renyi.hu/~szamuely/gal6-7.pdf. Unfortunately I don't know about the rest of this book. –  user717 Jul 13 '10 at 8:06

Nagata's Field Theory is an extremely deep book that a lot of my friends who are algebraic number theorists like immensely.

The most complete text I know on the subject is Patrick Morandi's Field And Galois Theory-it's also one of the most gentle and readable.

And of course,there's always the beautiful lecture notes by Irving Kaplansky.

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Another great set of notes by H.W.Lenstra which discusses algebra in general, and in particular Field and Galois theory at the end, can be found here (pdf). Two quotes about the approach:

We indulge next in a casual and motivational comparison of the classical and modern approaches to Galois theory. In all current textbooks, Galois theory is studied using ﬁnite separable ﬁeld extensions L of a given base ﬁeld K. Our approach follows that of the Grothendieck formulation, in which the objects under consideration are ﬁnite étale K-algebras A. We now consider the relation between the two perspectives.

and

One can track Galois theory through the years from being a discussion of polynomials, to an exploration of splitting ﬁelds, and ﬁnally to the Grothendieck formulation that we have used in this unit.

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I have not actually read this book entirely but Hideyuki Matsumura's Commutative Algebra is a relatively advanced text on the subject. I did read the first few chapters of this book, but having done so I prefer David Eisenbud's Commutative Algebra; in any case, there are certain important concepts which Matsumura discusses towards the end of his book which may be worthwhile to read. (Matsumura does occassionally allude to geometric connections in his book, but Eisenbud alludes to them far more often and in far greater depth.)

All that said, this text due to Matsumura, especially part 2 (the last four chapters) does have some more "advanced field theory" in the context of commutative algebra and algebraic geometry. On the other hand, Matsumura's other book (which I believe was published later) on Commutative Ring Theory also has some more advanced field theory towards the end of the book, and perhaps could be more useful since it is actually designed as a textbook in the subject. (Whereas, I believe, Matsumura's Commutative Algebra was not written with this as the primary goal.)

I should add, however, that it takes time to become accustomed to Matsumura's exposition. He does state several facts without proofs (and some of them are quite fundamental to the rest of the text) but if you are fairly accustomed to homological algebra and commutative algebra in general, you should have little or no difficulty working out the proofs yourself. Also, the prerequisites for both texts is "graduate-level algebra", the fundamentals of homological algebra (i.e., all homological algebra up to, and including, the development of the torsion and extension functors), and familiarity with the exterior algebra. The appendices in Matsumura's Commutative Ring Theory do give a description of the necessary background but are perhaps too condensed if you are not already familiar with the material.

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One the other hand, if you can read French (and know the vocabulary necessary to read mathematics books in French), you can always read Grothendiek's EGA but that might take you too far afield unless you wish to pursue algebraic geometry. (Out of interest, I have always wondered whether there is an English copy available of the EGA. Does anyone know whether there is, and if so, where I can find it? If not, are there English texts that cover similar material to the EGA that you would recommend?) –  Amitesh Datta Jul 13 '10 at 1:27
There have been no new editions of EGA (and certainly no English ones) for several decades, since Grothendieck refuses to give permission for such. –  Charles Staats Jul 13 '10 at 1:38
@Amitesh: I don't think that you address the specific question. –  Martin Brandenburg Jul 13 '10 at 12:38
Perhaps my answer was too long. Would this be a better answer? "The last few chapters of Matsumura's Commutative Algebra (i.e., part two of this text) do discuss advanced field theory especially in the context of commutative algebra and algebraic geometry." –  Amitesh Datta Jul 13 '10 at 13:28
This answers a question "what are commutative algebra books would you recommend", but Galois theory is not commutative algebra. –  Victor Protsak Jul 18 '10 at 13:40