On the factorization of the relative class number in terms of Frobenius divisions

The relative class number of an imaginary abelian number fieldK is—up to trivial factors—the product of the first Bernoulli numbersB_x belonging to the odd characters ofK. This product splits into rational factorsF_Z = {B; Z}, whereZ runs through the Frobenius divisions of odd characters. It is shown that each numberF_z is—up to a certain prime power—the index of two explicitly given subgroups of (K, +). These subgroups are cyclic Galois modules, whose generators arise from roots of unity and cotangent numbers, resp. Our result is an analogue of a result concerningh⁺ which was given by Leopoldt many years ago.To the memory of my friend Kurt Dietrich