Talk:Class field theory

Expansion

It appears that this page is in severe need of expansion, both on the main points, and on certain important complementary details. Since I am currently learning Class Field Theory, I thought I could make it a goal of mine to help expand the page on the subject in order to test my knowledge. So far, my vision of a more comprehensive CFT page includes sections for the following:

Formulation of (Global) Class field theory: Personally, I would like to completely rewrite the section that corresponds to this one, in order to give those who know the basics of Algebraic Number Theory a complete description of the main theorems of Global Class Field Theory. This probably implies that the theorems would be formulated at first in terms of the ideal class group instead of the idele class group, but I think the latter formulation is necessary to include afterward. Essentially, this is the approach taken in J. S. Milne's notes on the subject, in the first chapter on Global CFT. The way in which I would like to structure this section is with a subsection for background (including maybe ramification and Frobenius elements, Kronecker-Weber, etc. and also finite and infinite primes, ray class groups, etc.) and then one subsection for each main theorem (Artin Reciprocity, Existence Theorem, Chebotarev Density Theorem, etc.). Each theorem would be generalized to the point at which they describe ramified extensions, i.e. use moduli and ray class groups/fields. Finally, as mentioned above, it would be nice to give the idelic formulation of these points.

Local CFT: There is an article on Local CFT already, but it is a stub and I think it should be merged with this one. And in the end, it would be good to have a section structured similarly to the one above, but for Local CFT.

Approaches to CFT: I would like to have a section on the ways that the subject is approached, i.e. descriptions of the methods with which the main theorems are proved (examples below), and the philosophy behind them. Similar to the above sections, subsections for each approach would be nice. The following are examples of some approaches which would probably warrant their own subsections: Group Cohomology (Cf. Milne); Lubin-Tate Theory and formal group laws (For Local CFT) (Milne again); An abstract axiomatic approach (Cf. Neukirch).

Motivation and Importance: This is self-explanatory. A good source for ideas here is the text of Marcus. It only covers global theory though, and a large emphasis seems to be put on density theorems. It would also be nice to include a paragraph or two on the importance of finding a non-abelian Class Field Theory, and the relationship of this goal to the Langlands Program (this is briefly mentioned in the current version of the article too.)

Other CFT's: This section is best described by examples of its content: Explicit CFT: Kronecker's Jugendtraum, Hilbert's 12th problem, etc. Abstract CFT: This is an axiomatic Class Field Theory, presented in Neukirch. In proving the main theorems of local and global CFT, one may develop this and verify its axioms for local and global fields. Et Cetera.

I would like to add a compliment: It would be good to include in the section on generalizations a description of the "Geometric Class Field Theory" of Lang and Rosenlicht, exposed in Serre's text "Algebraic Groups and Class Fields" for one.

Any discussion with or against the above points is more than welcome. Kimbesque (talk) 23:31, 18 May 2013 (UTC)

The role of class field theory in algebraic number theory

This section seems rather overblown: "the key part and the heart of algebraic number theory"; "thousands of applications in number theory". Is there a reliable source for such assertions? Spectral sequence (talk) 18:07, 30 June 2013 (UTC)