Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I know the p-adic method is important in algebraic number theory. However, in the old days, the global class field theory was developed using only ideals and classical analysis. I'm curious to know about it. Another reason is that I think the ideal theoretic approach is more constructible than the p-adic one. Since those old books and papers were written in German and I'm not at all good at German, I prefer a book written in English. Is there such a book?

share|improve this question
I haven't read the original papers, so I don't know how closely they follow them, but Lang's "Algebraic Number Theory", Janusz's "Algebraic Number Fields", and Childress' "Class Field Theory" all take the "traditional" approach to CFT. –  B R Apr 24 '12 at 2:38
Would you also mind registering with the site? It helps the site keep better track of your questions; comments and so on. It is real easy--login with your favourite OpenID provider. –  user21436 Apr 24 '12 at 2:40
@BR Lang, Janusz, Childress all used the p-adic method. –  Makoto Kato Apr 24 '12 at 4:23
@MakotoKato, it depends on what you mean. None of them deduce global CFT from local CFT, which is what I thought you were trying to avoid. It is true that they use $p$-adic numbers in their proofs of the Second Inequality, but not in an essential way. You can replace their computations with purely global ones (in fact, see 11.3 and 11.4 of Lemmermeyer's book). –  B R Apr 24 '12 at 4:51
@BR "You can replace their computations with purely global ones" I knew this. My question was how one could do it. –  Makoto Kato Apr 24 '12 at 21:42

1 Answer 1

You could look at Lemmermeyer's book. (It doesn't include a proof of Artin's reciprocity law, but includes proofs of the first and second inequality, and has a lot historical backgroud.)

share|improve this answer
Thanks for the link. –  Vadim Apr 24 '12 at 3:35
@Matt Thanks! I'll check it. –  Makoto Kato Apr 24 '12 at 3:51

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.