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.