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.

**Edit**
Since Anon's answer and Franz' comments are contradicting, I started bounty.

2more comments