On the factorization of the relative class number in terms of Frobenius divisions (Q1314861)

Let \(K\) be an imaginary abelian number field of conductor \(n\), \(K\subseteq \mathbb{Q} (\zeta_ n) \subseteq \mathbb{C}\). Let \(G= \text{Gal} (K/\mathbb{Q})\). The class number \(h\) of \(K\) splits into \(h= h^ + \cdot h^ -\), where \(h^ +\) is the class number of \(K\cap \mathbb{R}\) and for some easily obtained constant \(C\), \(C\cdot h^ -= \prod | B_ \chi|\), where the product runs through \(X^ -\), the set of odd characters \(\chi\) of \(G\), and \(B_ \chi\) is the primitive first Bernoulli number of \(\chi\). The Frobenius division of \(X^ -\) is a partition of \(X^ -\) into classes \(Z= Z(\chi)\) where \(\psi\) is in \(Z(\chi)\) iff \(\langle \psi\rangle =\langle \chi\rangle\). The author shows that for each \(Z\), \(F_ Z= \prod_{\chi\in Z} | B_ \chi |\) is, up to an easy constant, the index of two explicitly given \(\mathbb{Z} G\)-cyclic submodules of the additive group of \(K\). The result is analogous to the ``much more involved'' decomposition of \(h^ +\) of \textit{H. W. Leopoldt} [Abh. Deutsch. Akad. Wiss. Berlin, Kl. Math. Naturwiss. 1953, No. 2 (1954; Zbl 0059.035)].