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?
Global class field theory without p-adic method
2
$\begingroup$
algebraic-number-theory
-
1I 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. – 2012-04-24
-
2Would 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. – 2012-04-24
-
0@BR Lang, Janusz, Childress all used the p-adic method. – 2012-04-24
-
0@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). – 2012-04-24
-
0@BR "You can replace their computations with purely global ones" I knew this. My question was *how* one could do it. – 2012-04-24
-
0@BR You misunderstand me. I know it 'cause Matt answered my question. I just replied to your comment. Thanks anyway. – 2012-04-24
-
0@BR Usually there can be several solutions to a mathematical problem. By accepting an answer, you shut out other possibly interesting or even better solutions? – 2012-04-26
-
0@BR That's good. But are you sure that accepting an answer has no negative side effect like discouraging people to add a new proof? – 2012-04-26
-
0@BR As I wrote, usually there are several(or many) solutions of a mathematical problem. A problem of accepting an answer is that we may lose a chance that someone might add a new and interesting or even better answer. – 2012-05-21
-
0@MakotoKato, I removed comments that were no longer relevant. – 2012-05-22