Central simple algebras approach to class field theory, merits of

Question

As noted earlier, I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly rigorous and clean.

In any case, the second part of the book is "thoroughly unmodern", in the author's own terms. He dispenses with cohomology and builds the whole theory based on central simple algebras. And he advises the reader to make it an exercise to himself for working out the "hidden cohomology" in it.

I personally felt this as some kind of perversion(just as in the fixation for Haar measure in the first part) to show the reader that he is a powerful mathematician could do things precisely the way he wants, and my feeling was strengthened by the title "Basic Number Theory".

However since he is a great mathematician, and I am nobody, I feel unqualified dismiss him like that. There must be some merits/uses of doing it this way, rather than using cohomology. If somebody can enlighten me, I would be very grateful.

Again, is it the case that class field theory can be done with just $H^1$? I wanted to study the subject and attempted, but was put off for a long time by the wrong book choice, reading Weil which felt like drinking stone soup, and it takes me time to take up the subject again.

Answer 1

If you are looking for a more conceptual way of understanding the central simple algebras approach to class field theory, I think that you would be well advised to look at Roquette's book The Brauer-Hasse-Noether Theorem in Historical Perspective. The Brauer-Hasse-Noether Theorem, oftentimes referred to as the Albert-Brauer-Hasse-Noether theorem (Adrian Albert, an american, discovered part of the theorem independently), states that a central simple algebra defined over a number field splits globally if and only if it splits everywhere locally and is intimately connected with class field theory.

The entire book is only 80 pages long, and takes only a few hours to read. You should especially look at Chapter 6: The Brauer group and class field theory. The last few pages of this chapter make explicit the application of work done with central simple algebras to class field theory and the connection between this approach and the more modern cohomological approach.

[Edit 1] As Franz has pointed out, applying the theory of central simple algebras to class field theory is not just another approach, but rather was the original approach. The connection with cohomology arises as follows:

For a field extension $K/k$, denote by $Br(K/k)$ the Relative Brauer group of $k$ with respect to $K$. Explicitly, it is the kernel of the homomorphism $Br(k) \rightarrow Br(K)$. Then $Br(k)=\cup Br(K/k)$. It isn't too hard to show that the elements of these relative Brauer groups (for $K/k$ Galois) are in one to one correspondence with certain factor sets relative to $K$. These factor sets naturally arise as the elements of $H^2(K/k)$, ultimately yielding an isomorphism $Br(K/k)\cong H^2(K/k)$. This isomorphism allows one to translate results like the Albert-Brauer-Hasse-Noether theorem into the perhaps more familiar cohomological language.

[Edit 2] Apparently Roquette's book is available online (along with many other gems).

Answer 2

Weil's approach is the historical one: during the early 1930s, Hasse (with the help of a lot of people too numerous to mention) succeeded in building up class field theory using central simple algebras. Later, in the 1950s, people realized that for getting class field theory, the theory of algebras could be replaced by the formalism of cohomology: Brauer groups, for example, which had a very concrete meaning, turned into abstract cohomology groups. For Hasse, and perhaps also for Weil (?), the purely cohomological approach was "all bones, no meat" - see Hasse's comments in his history of class field theory in Cassels - Fröhlich.

Personally, I'd rather read the dry cohomological version (for example in Neukirch's "Klassenkörpertheorie", which has a good chance of being translated into English soon) than Weil's book for pretty much the same reasons as you. In this book, Neukirch builds up a full grown version of cohomology for finite abelian groups (I always liked the book by Weiss - Cohomology of Groups - as a first introduction to the stuff you need for class field theory), whereas in his later book he manages to do everything with just H⁰ and H¹ (and profinite groups).

Introductions to class field theory using algebras are rare; the book Algebra II (Springer 2008; Engl transl. by S. Levy) contains an introduction to local class field theory via algebras.