Computation of class numbers of number fields
Let \(K/k\) be an extension of number fields. Assume that the quotient \(\zeta_K/ \zeta_k\) of the Dedekind zeta-function of \(K\) and that of \(k\) is homomorphic in the whole complex plane. Note that this assumption holds as soon as \(K/k\) is normal. We give an expression for the value \((\zeta_K/ \zeta_k) (1)\) as the limit of a rapidly convergent series, which makes it convenient to numerically compute this value \((\zeta_K/ \zeta_k) (1)\). Our main purpose is to compute relative class numbers of non-abelian CM-fields \(K\), taking \(k\) as the maximal totally real subfield of \(K\) (here \(K/k\) is quadratic, thus normal). For example, we explain in detail how to compute the relative class number of a non-normal quartic CM-field, and provide tables of relative class numbers of such CM-fields. The reader may find other illustrations by example of the method developed in this paper in the following two papers [`Calcul des nombres de classes relatifs: applications aux corps octiques quaternioniques à multiplication complexe', C. R. Acad. Sci., Paris, Sér. I 317, 643-646 (1993; Zbl 0795.11059), and `Calcul des nombres de classes relatifs de certains corps de classes de Hilbert', ibid. 319, 321-325 (1994; Zbl 0817.11049)].
scientific article; zbMATH DE number 912155