Central simple algebras approach to classfield theory, Merits of.

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 classfield 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.

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The theory of central simple algebras is a really beautiful theory. – Harry Gindi Feb 5 '10 at 16:05
Oh I have no doubt. But cohomology is more modern and much more intricate, and applied to various settings. – Anweshi Feb 5 '10 at 16:08
"is it the case that classfield theory can be done with just H^1?". No! One of the fundamental constructions of class field theory is the fundamental class, which is in an H^2. Moreover, cupping with it and hence moving from H^r to H^{r+2} is explicitly used for r=-2,-1,0 (use Tate cohomology in negative degrees). Furthermore implicit in some constructions is the notion of "dimension shifting" which might require other degrees. There are examples of where cohomology groups other than H^1 are used. – Kevin Buzzard Feb 5 '10 at 16:22
Dear Kevin, One should remember though tha Takagi proved CFT without (explicit) cohomology, and Artin proved his reciprocity law in the same way. I don't think it is a question of H^1 vs. all the H^i, but rather all cohomology or none. – Emerton Feb 5 '10 at 17:18
Kevin: Perhaps I got you wrong but Neukirch's approach to CFT (which yields the same reciprocity morphisms as the cohomological approach) does not use cohomology. Of course this approach does not provide cohomological information at first, that's right. – user717 Feb 5 '10 at 19:08

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).

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Thanks for this reference, which looks great. (I've only just skimmed a few pages so far, and already learnt many interesting things!) – Emerton Feb 5 '10 at 19:35
Agreed, that book looks nice, I have been looking for something like that for a while. Thanks. – Hailong Dao Feb 5 '10 at 23:55
It is such a great pity that I cannot add a second upvote for your edit 2. – Anweshi Feb 6 '10 at 1:46

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.

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Anyone who has learned class field theory without at least seeing the Brauer group is missing out. – Harry Gindi Feb 5 '10 at 16:32
But, what are the motivations of Weil to take such an unmodern approach? The older approach would have been available in older references. Why did he decided to take it up in "Basic Number Theory", where in the first part he took up a complete new approach using Haar measure right, left and center? – Anweshi Feb 5 '10 at 16:39
Neukirch's "Class Field Theory" exists already in English. I have it on my bookshelf. – Felipe Voloch Feb 5 '10 at 17:08
Sorry for not having been precise enough - "Class field theory" is his second book; the first one in German only has the title in common with this one. @Anweshi: I prefer "classical" to "unmodern". Weil grew up (mathematically) in the 1930s, when the approach via algebras was modern. I perfectly understand his motivation for promoting the Haar measure approach: Weil had written a book on integration ("L'intégration dans les groupes topologiques et ses applications) in 1940. – Franz Lemmermeyer Feb 5 '10 at 17:45
@Harry: I think you're absolutely right. – user717 Feb 5 '10 at 19:12