An advanced exposition of Galois theory

Question

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?

Answer 1

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 .

Answer 2

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

Answer 3

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.

Answer 4

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.

Answer 5

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.