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I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about global class field theory from the corresponding local facts. This often means that global class field theory is given the idelic formulation, as local fields have already been covered.

Alternatively, I'm about to take a course on class field theory (which is the sequel to an undergraduate course on algebraic number theory and basic zeta/L-functions) which dives directly into global class field theory and will follow the original (1920s) formulations (ideal-theoretic) and proofs of the basic results.

I'm wondering what are people's opinions of the two different approaches to class field theory. Does it make more sense to start local and go global, or is it a better idea to learn the subject more historically? I asked my professor here at Princeton about it, since I was aware that Harvard's CFT course starts with local, he responded that since what we're really interested in are number fields anyway, it's much more relevant to proceed immediately with global class field theory. Thoughts?

EDIT/UPDATE: Based on input from this thread and more experience, here is the approach I've decided to follow:

  1. Learn global class field theory using more elementary proofs, following something like Janusz (or another source if you don't like Janusz's style)

  2. Learn the cohomology-heavy proofs of local class field theory. I particularly like Milne's notes for this.

  3. Continue and learn the proof of global class field theory using cohomology of ideles. You could just continue in Milne, or try the chapters in Cassels-Frohlich

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I'll just remark that history is repeating itself: the fairly recent proof by Harris and Taylor of local Langlands for GL_n is a global proof, and as far as I know no local proof is currently known. –  Kevin Buzzard Nov 27 '09 at 13:15
    
I should add that the approach I've decided to use is the following, and I recommend that others try it: 1. Learn global class field theory using more elementary proofs, following something like Janusz (or another source if you don't like Janusz's style) 2. Learn the cohomology-heavy proofs of local class field theory. I particularly like Milne's notes for this. 3. Continue and learn the proof of global class field theory using cohomology of ideles. You could just continue in Milne, or try the chapters in Cassels-Frohlich. –  David Corwin Aug 23 '10 at 2:58
    
I looked at several books on CFT. The only one I feel I understood, at least partially, is Weil's "Basic Number Theory". I find Weil's writing very clear. The book is almost selfcontained. The main prerequisite is the theory of locally compact abelian groups. I believe this is unavoidable, but I'm not sure. At any rate, this makes things more transparent. [Weil's approach is a local to global one.] –  Pierre-Yves Gaillard Aug 23 '10 at 5:14
    
@Pierre-Yves: You really don't need the theory of locally compact abelian groups to learn class field theory. It might be true that that book is great if you have that prerequisite, but I imagine that many who want to learn class field theory will not have that prerequisite. –  David Corwin Aug 23 '10 at 13:35
    
Dear Davidac897: Thanks for your answer. I don't doubt that you're right, but I'll still try to make the following point. CFT is about loc. cpt ab. gps (LCAG): adele rings, idele groups, Galois groups, are LCAG, and the morphisms between them are continuous, that is LCAG morphisms. Also, the theorems about LCAG needed to read the book are very clearly stated by Weil, and very easy to understand (as far as their statement is concerned). You can take them for granted. [Why is the adele ring defined as a restricted product? To make it locally compact.] –  Pierre-Yves Gaillard Aug 24 '10 at 5:10
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9 Answers 9

I learned class field theory from the Harvard two-semester algebraic number theory sequence that Davidac897 alluded to, so I can really only speak for the "local first" approach (I don't even know what a good book to follow for doing the other approach would be, although I found this interesting book review which seems relevant to the topic at hand.).

This is a tough question to answer, partly because local-first/global-first is not the only pedagogical decision that needs to be made when teaching/learning class field theory, but more importantly because the answer depends upon what you want to get out of the experience of learning class field theory (of course, it also depends upon what you already know). Class field theory is a large subject and it is quite easy to lose the forest for the trees (not that this is necessarily a bad thing; the trees are quite interesting in their own right). Here are a number of different things one might want to get out of a course in class field theory, in no particular order (note that this list is probably a bit biased based on my own experience).

(a) a working knowledge of the important results of (global) class field theory and ability to apply them to relevant situations. This is more or less independent of the items below, since one doesn't need to understand the proofs of the results in order to apply them. I second Pete Clark's recommendation of Cox's book /Primes of the form x^2 + ny^2/.

Now on to stuff involved in the proofs of class field theory:

(b) understanding of the structure and basic properties of local fields and adelic/idelic stuff (not class field theory itself, but material that might be taught in a course covering class field theory if it isn't assumed as a prerequisite).

(c) knowledge of the machinery and techniques of group cohomology/Galois cohomology, or of the algebraic techniques used in non-cohomology proofs of class field theory. Most of the "modern" local-first presentations of local class field theory use the language of Galois cohomology. (It's not necessary, though; one can do all the algebra involved without cohomology. The cohomology is helpful in organizing the information involved, but may seem like a bit much of a sledgehammer to people with less background in homological algebra.)

(d) understanding of local class field theory and the proofs of the results involved (usually via Galois cohomology of local fields) as done, e.g. in Serre's /Local Fields/.

(e) understanding of class formations, that is, the underlying algebraic/axiomatic structure that is common to local and global class field theory. (Read the Wikipedia page on "class formations" for a good overview.) In both cases the main results of class field theory follow more or less from the axioms of class formations; the main thing that makes the results of global class field theory harder to prove than the local version is that in the global case it is substantially harder to prove that the class formation axioms are in fact satisfied.

(f) understanding the proofs of the "hard parts" of global class field theory. Depending upon one's approach, these proofs may be analytic or algebraic (historically, the analytic proofs came first, which presumably means they were easier to find). If you go the analytic route, you also get:

(g) understanding of L-functions and their connection to class field theory (Chebotarev density and its proof may come in here). This is the point I know the least about, so I won't say anything more.

There are a couple more topics I can think of that, though not necessary to a course covering class field theory, might come up (and did in the courses I took):

(h) connections with the Brauer group (typically done via Galois cohomology).

(i) examples of explicit class field theory: in the local case this would be via Lubin-Tate formal groups, and in the global case with an imaginary quadratic base field this would be via the theory of elliptic curves with complex multiplication (j-invariants and elliptic functions; Cox's book mentioned above is a good reference for this).

Obviously, this is a lot, and no one is going to master all these in a first course; although in theory my two-semester sequence covered all this, I feel that the main things I got out of it were (c), (d), (e), (h), and (i). (I already knew (b), I acquired (a) more from doing research related to class field theory before and after taking the course, and (f) and (g) I never really learned that well). A more historically-oriented course of the type you mention would probably cover (a), (f), and (g) better, while bypassing (b-e).

Which of these one prefers depends a lot on what sort of mathematics one is interested in. If one's main goal is to be able to use class field theory as in (a), one can just read Cox's book or a similar treatment and skip the local class field theory. Algebraically inclined people will find the cohomology in items (c) and (d) worth learning for its own sake, and they will find it simpler to deal with the local case first. Likewise, people who prefer analytic number theory or the study of L-functions in general will probably prefer the insights they get from going via (g).

I'm not sure I'm reaching a conclusion here: I guess what I mean to say is -- I took the "modern" local-first, Galois cohomology route (where by "modern" we actually mean "developed by Artin and Tate in the 50's") and, being definitely the algebraic type, I enjoyed what I learned, but still felt like I didn't have a good grip on the big picture. (Note: I learned the material out of Cassels and Frohlich mostly, but if I had to choose a book for someone interested in taking the local-first route I'd probably suggest Neukirch's /Algebraic Number Theory/ instead.) Other approaches may give a better view of the big picture, but it can be hard to keep an eye on the big picture when going through the gory details of proving everything.

(PS, directed at the poster, whom I know personally: David, if you're interested in advice geared towards your specific situation, you should of course feel welcome to contact me directly about it.)

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Alison, the link to a book review in your first paragraph is broken and it's not clear from your text what book it is, so only you can fix that. –  KConrad Aug 23 '10 at 0:15
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Edited to fix the link. For further reference, in case the link changes again, it was to an maa.org review, written by Gouvea, of Nancy Childress's Class Field Theory. –  Alison Miller Aug 23 '10 at 0:39
    
@Alison Miller: Could you please elaborate on what you said: "I learned the material out of Cassels and Frohlich mostly, but if I had to choose a book for someone interested in taking the local-first route I'd probably suggest Neukirch's /Algebraic Number Theory/ instead."? Why do you think Neukirch is a better choice? –  Brian Apr 7 '11 at 2:49
    
(caveat: I've read neither book in its entirety, so this is based on only having read parts of both, and for different purposes) I find Neukirch to be less dense, more elegant, and generally a joy to read. Cassels and Frolich is lecture notes from an instructional conference, which means it has different chapters by different people, and is generally a bit rougher around the edges (in some ways this is a good thing of course). –  Alison Miller Apr 7 '11 at 3:49
    
@Alison Miller: Thanks! –  Brian Apr 7 '11 at 16:10
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There is no royal road to class field theory -- to understand it well is going to take lots of time and multiple exposures no matter what.

That said, when covering this material in courses at UGA I have had some success with the following approach: first discuss the statements of global class field theory in the classical ideal-theoretic language, and give some motivation for these results. For instance, Cox's book Primes of the Form x^2 + ny^2 is good for this: some lecture notes for a course based on Cox's book are available at

http://www.math.uga.edu/~pete/primesoftheform.html

Then I would recommend studying local class field theory from the perspective of Galois cohomology. For this, Serre's book Local Fields is still a classic; Jim Milne has some very nice lecture notes as well.

Only then would I venture into the realm of idele-theoretic global class field theory. But again, these are just my two cents.

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Yes, as you correctly describe it, there are two main approaches to class field theory, the classical (1920s) approach in terms of ideals,and the later (Chevalley-Artin-Tate) approach in terms of ideles and cohomology. The first takes you to the main theorems more quickly and easily, but the second gives you much more. Fortunately, they are not incompatible, so learning the classical approach will be a big help if you then decide to learn the second approach.

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As many people have indicated above, class field theory is large and difficult, and no approach is going to make it easy.

My personal experience was that it was crucial to understand the statements of the main theorems of class field theory well before learning any of the proofs. I tried to learn class field theory from many books and teachers before succeeding, and I think this is what made everything click for me. To my mind, the results of class field theory are a beautiful cohesive whole. It is often easy to see how they are consistent with and partially imply each other, while seeing why any one of them is true is very difficult.

For this purpose, I would suggest learning the global statements before the local ones, because they are more elementary and because you probably have more experience with extensions of number fields than with extensions of local fields. I don't think it matters so much what order you learn the proofs in.

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Perhaps no one else will share my opinion but I am a fan of Neukirch's approach to both local and global class field theory as presented in his book on algebraic number theory. Neukirch constructs an abstract framework including several objects and conditions which then induce a concept of a class field theory. All this is modeled upon the situation of local fields and the correspondence between prime elements and Frobenius automorphisms for unramified extensions. Hence, one has a nice motivation for this approach and moreover it is pretty elementary (although I have to admit that the verification of the multiplicativity of the reciprocity morphism is "dirty" but one can just believe this and save some time). In particular, group cohomology is not used. Neukirch then shows how to really get local class field theory from this abstract approach. The verification of the conditions mentioned above is not that hard (the existence theorem requires additional work). So, I think that this is a great path to the general concept of class field theories with local class field theory being the first example and motivation. The point is that from the same abstract framework one can also get global class field theory. This is unfortunately more technical but I think that this also provides a lot of insight.

For me the cohomological approach via Nakayama-Tate duality was always a mystery. I think if one does not learn how group cohomology appeared implicitly via the algebra theoretic considerations in class field theory, then it will remain to be a mystery. But this may just be a result of my lack of knowledge...

One could remark that a drawback of Neukirch's approach is that one does not get information about higher cohomology groups which is important elsewhere. But as Neukirch's class field axiom implies that the discrete module under consideration gives a class formation, I am not sure if this is really true...

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My experience was similar to that of David Speyer. My first encounter with class field theory was the very motivating "Introduction to the construction of class fields" by H. Cohn. That book motivates through questions on rational triangles, primes represented by quadratic forms, analogies from Riemann surfaces and the use of modular forms, Klein's icosahedron. An other very motivating aspect of it is the topological idea behind Kato's generalization, describing $\pi_1^{ab}$ of a scheme in different ways, and the geometry associated with that and ${\mathbb{F}}_1$-yoga. Conc. the cohomological approach, my impression was opposite to that of Arminius.

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There are three language for class field theory,the first is analytic,such as Iwasawa's book,second is using ideal or idele,such as Niekrich's book,the last is more powerful--cohomolgy,such as Milne's notes on his homepage. My experience was local to global using cohomolgy.

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I think Iwasawa's book"Local class field" is the best one for you.If you cannot find it in your library,maybe I can send a e-vision to you.My email is: where189@yahoo.com.cn.

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Does anyone have any idea where I could find a book or notes which develop class field theory using the elementary analytic methods?

EDIT: See Reference for Learning Global Class Field Theory Using the Original Analytic Proofs? for the answers to this question.

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It would be most appropriate to post this as a question in its own right (or at least as an edit to the original question above), and you're likely to get more answers that way. –  Alison Miller Dec 6 '09 at 21:09
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