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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?2012-04-24
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    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!2012-04-26
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    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.2012-04-26
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    @Dylan you still misunderstand me. I know almost all the modern proofs using local fields, ideles, cohomology. I asked the question because as I wrote ideals are more concrete and elementary, in other words, down to earth.2012-04-26
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    I guess this question is more suitable in MO so I posted it there.2012-04-26
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    [The MO question](http://mathoverflow.net/questions/95225/)2012-04-26
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    @MakotoKato, I was wondering if you thought Lemmermeyer's book that Matt E linked to (http://math.stackexchange.com/a/136076/5773) did not do this. Does he use methods you want to avoid, or did you find his argument otherwise unsatisfying (it is still a draft)?2012-04-26
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    @BR Lemmermeyer's book uses the classical method, i.e. L-functions. In his book he called it the first inequality(this is classical usage).2012-04-26
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    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.2012-04-26
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    @Will What if I can't understand an answer for long time?2012-04-27
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    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.2012-04-27
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    @Will I asked 17 questions in ME. 10 were ununswered. I accepted 3. One is obviously wrong(Fubini). One is far from a final solution(Are computers ...) There remain only two.2012-04-27
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    @Will Suppose there is a *purely* mathematical question. People answer it. Does it really matter whether a questioner is satisfied with an answer or not? I think what matters is whether a question is clear and useful or not, and whether an answer is correct or not.2012-04-27
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    @Will Are you implying one should not ask a question which is ahead of one's current level?2012-04-27
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    @MakotoKato I'm able to find scanned copies of Iyanaga's book online with a little bit of effort. I won't besmirch the reputation of this fine website by posting a link, but hopefully you can grab a copy as well.2012-04-27
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    It appears that one cannot view the Iyanaga notes online for free. The book can be purchased, http://www.amazon.ca/Class-Field-Mathematics-Professor-Iyanaga-University/dp/B0007DM1YS and is not expensive2012-05-01
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    @Will Thanks, I ordered it. It'd be nice for someone else to read the book and check it, though.2012-05-01
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    @Brett Suppose there is a PURELY mathematical question. Does it matter whether the questioner is satisfied with an answer or not? I got impression that there are too many REPUTATION HUNGRY people in this forum. I hope it's false impression.2012-05-04

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