A theorem of Hecke asserts
that the discriminant dK∕F of an extension of algebraic number fields K∕F a square
in the absolute class group. In 1932 Herbrand conjectured the following related
theorem and was able to prove it for metacyclic extensions: If K∕F is normal, then
dK∕F can be written in the form A2(𝜃), 𝜃 ∈ F; where (i) 𝜃 ≡ 1 (mod)B, B is the
greatest divisor of 4 which is prime to dK∕F, and (ii) 𝜃 > 0 at each real prime v
except when K ⊗FFv is a direct sum of copies of the complex fleld and (K : F) ≡ 2(mod4).
More recently, A. Fröhlich gave a unified treatment of these and related
questions using the concept of an idèle discriminant. The purpose of this paper is to
present a generalization of these results with some connections with the structure of
the Galois group.
