Formulae for the relative class number of an imaginary abelian field in the form of a determinant (Q2764674)

Let \(K\) be an imaginary abelian number field with conductor \(m\) and Galois group \(G = \text{Gal} (K/ \mathbb Q)\). Since the result of \textit{L. Carlitz} and \textit{F. R. Olson} [Proc. Am. Math. Soc. 6, 265-269 (1955; Zbl 0065.02703)] on Maillet's determinant, many papers investigated matrices over \(\mathbb Q\), whose determinant equals -- up to a more or less known factor -- \(h_K^-\), the minus part of the class number of \(K\). Usually, these matrices can be interpreted as transformations of lattices within the (minus part of the) rational group ring over \(G\). The characters of \(G\), being identified with Dirichlet characters, yield an orthogonal decomposition of the group ring and a factorization of the determinant into ``generalized'' Bernoulli numbers, which are connected to \(h_K^-\) by the analytic class number formula. NEWLINENEWLINENEWLINEThis paper starts with an overview of known results concerning Maillet's determinant and Demyanenko's matrix, which all can be obtained by specializing Theorem~1 of this paper, giving a formula for a very general determinant. More precisely, let \(\theta_n' \in \mathbb Q [G]\) be the Stickelberger element coming from level \(n \mid m\) and \(\theta\) be a linear combination of these elements with arbitrary coefficients from \(\mathbb Q [G]\). Then the determinant of Theorem~1 is essentially the index of the ideal generated by \(\theta\) within the imaginary part of the integral group ring over \(G\).