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

I posted this question in Stack Exchange, but got no answer nor positive vote. So I crosspost this here.

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 local fields, 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?

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

Edit Thanks, Masato. 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'd like to start a bounty on this question. How can I do it?

Edit At least two opposite answers appeared so far. Franz Lemmermeyer wrote no one has found such a proof and Iyanaga uses (non-archimedian) local fields everywhere in his proof, while Anon wrote otherwise. I wonder which is correct.

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The math.SE question: math.stackexchange.com/q/136456/5363 –  Theo Buehler Apr 26 '12 at 4:26
What does "there is" mean? So far, no one has found such a proof, and I guess that if one is ever found, it will be even more technical than an elementary proof of the prime number theorem. –  Franz Lemmermeyer Apr 26 '12 at 8:55
@Franz Judging from what Iyanaga wrote in his book "The theory of numbers", it seems that his proof which is based on the idea of the Chevalley's proof is not so much a feat. So I guess it's not so technical as an elementary proof of the prime number theorem unless you call the Chevalley's proof so. – –  Makoto Kato Apr 29 '12 at 23:52
Chevalley introduced the notion of ideles to study infinite dimensional extentions of algebraic number fields. Iyanaga wrote that ideles were not indispensable in the Chevalley's proof of the second inequality contrary to the common and unfounded belief. He said that the Chevalley's proof depended on a lemma which can be stated and proved using only ideals(and infinite prime spots). –  Makoto Kato Apr 30 '12 at 7:00
Makoto, regarding bounties, see here: mathoverflow.net/faq#bounty –  B R May 1 '12 at 4:20

I heard that Iyanaga's class field notes had such a proof. See also his article on the history of class field in his book the theory of numbers.

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This is actually correct in that Iyanaga gives Chevalley's proof without using ideles. But of course he cannot avoid the whole local-global machinery, and in particular he uses local fields everywhere. –  Franz Lemmermeyer Apr 26 '12 at 17:20
@Franz Since I don't have Iyanaga's class field notes at hand, I cannot see his proof at present. However, he wrote that he proved the second inequality utilizing only the classical terms of the ideal theory in his book the theory of numbers(p.507). Is this his misconception? –  Masato Apr 26 '12 at 22:08
One man's classical term is another man's advanced notion. Iyanaga's notes start with a chapter on valuations, and he uses local fields. It is of course easy to eliminate them by replacing every p-adic argument by one on sequences of congruences. But as I said, this does not at all add clarity. –  Franz Lemmermeyer Apr 27 '12 at 17:36
@Franz Iyanaga wrote he proved the second inequality utilizing only the ideal theory. –  Makoto Kato Apr 29 '12 at 21:47
And I wrote what I wrote. Get the book. Read it. –  Franz Lemmermeyer May 1 '12 at 5:59
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Having looked at Iyanaga's notes, I'd say that Masato is correct: Iyanaga proves the main theorems of class field theory without using analysis, local class field theory, ideles, or cohomology (in any serious way).

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@Anon Thanks. I'd like to hear Franz' comment on your answer before I accept it. –  Makoto Kato May 5 '12 at 23:37
Perhaps I'm wrong, but I find it rude when someone asks me repeatedly whether I am able to read, or whether I know what I wrote. And when Iyanaga uses the exponential function in local fields, he is, as far as I am concerned, using p-adics and analysis. –  Franz Lemmermeyer May 7 '12 at 7:46
Makoto, you seem to be misunderstanding. Franz was saying that he thinks you are being rude, but that other people may disagree. For the record, I think that on the internet it is easy to sound more rude than you intend. A good way to counteract this is to try to act more polite than you intend (something I try to implement but often forget). –  B R May 9 '12 at 14:21
@BR I guess you are right. :-) –  Makoto Kato May 9 '12 at 15:11
I really do not like to barge in on matters like this, but to me Makoto's comment seems civil enough. On the other hand, BR's advice is more than sound :) –  Felix Goldberg May 14 '12 at 10:38