# Ideal theoretic proof of the second inequality of global class field theory

Classically the second(or the first in the old terminology) inequality of global class field theory($≦ [L : K]$, see, for example, the Milne's course note) was proved using Zeta functions and L functions. Modern proofs use ideles and group cohomology. Is there a proof of the second inequality using only ideals(i.e. no p-adics, no ideles, no analysis) and preferably no cohomology?

EDIT Ideals of algebraic number fields are more concrete and elementary than ideles. So I think this question is not uninteresting.

EDIT Iyanaga wrote, in his book "The theory of numbers" (p.507), that he proved the second inequality utilizing only the classical terms of the ideal theory in his "Class field theory, Chicago Univ. 1961". Could anyone please confirm this?

EDIT I crossposted this question in MO.

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It's sometimes(or more ofen than otherwise) nice to have several answers each of which has its own merit. Another reason is that I don't want to accept an answer which I don't understand fully(this is often my problem). It takes time to understand something fully. And usually there can be several solutions to a mathematical problem. By accepting an answer, you shut out other possibly interesting or even better solutions? – Makoto Kato Apr 24 '12 at 23:43
I think we're talking past each other as well. I did see that you knew that $L$-series could be used in the proofs, but in recent accounts I often see proofs which marry these with the newer tools of local fields, ideles, and cohomology. At least, this is how I learned the theorems. My (incorrect) impression from the end of the first paragraph was that your goal was to avoid these last things in particular, so I suggested relatively modern accounts that did so. I was not about to suggest that you go back and read Takagi! – Dylan Moreland Apr 26 '12 at 2:47
If you want to avoid analysis as well, then I'm afraid that I don't know of any "fourth path". I hope someone can say something more definitive. – Dylan Moreland Apr 26 '12 at 2:47
Putting this and Franz's skeptical answer on MO together with some 17 questions in 22 days, I suggest that you wait to ask a new question until you have understood the answers from the previous one, and that you answer as many questions as you ask. Also, you have not registered or given any information about your background. – Will Jagy Apr 26 '12 at 16:44
In that case, you have a clear task ahead of you, finding out background. Asking more questions that are too far ahead of your current level of training is something of a waste of the effort of other people. Meanwhile, as far as finding out what does make for an appropriate question, try answering a number of the basic ones here. After a relatively small number of answers, you will have a much better sense of what the questioner owes to the people who might choose to answer. – Will Jagy Apr 27 '12 at 1:47