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Formulae for the relative class number of an imaginary abelian field in the form of a determinant. (English) Zbl 1002.11079
Let $$K$$ be an imaginary abelian number field with conductor $$m$$ and Galois group $$G = \text{Gal} (K/ \mathbb Q)$$. Since the result of L. Carlitz and 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.
This 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 $$G$$.
Reviewer: G.Lettl (Graz)

##### MSC:
 11R29 Class numbers, class groups, discriminants 11R18 Cyclotomic extensions 11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
Zbl 0065.02703
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