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Ideal class groups of cyclotomic number fields. I. (English) Zbl 0837.11059

Rather than just cyclotomic fields, this article concerns more general CM-fields, i.e., totally complex quadratic extensions of totally real number fields. Let \(K\) be such a CM-field over a totally real field \(K^+\). The author first discusses the unit index in \(K\) (in Hasse’s sense), providing a unified treatment for its known properties and generalizing several results from the case of abelian fields. Secondly he proves that if \(K\) is contained in another CM-field \(L\), then its relative class number \(h^- (K)\) divides a product of the form \(h^- (L) \cdot 2^t\), where the latter factor depends upon the capitulation in \(L\) of the ideal classes of \(L^+\) and upon the Hilbert class fields of \(K\) and \(K^+\). In particular, if \(K\) and \(L\) are (full) cyclotomic fields, then \(t = 0\) and the result reduces to one proved by J. Masley and H. Montgomery [J. Reine Angew. Math. 286/287, 248-256 (1976; Zbl 0335.12013)]. Finally the author studies a formula expressing the relative class number of the composite of two CM-fields \(L_1\) and \(L_2\) in terms of \(h^- (L_1)\) and \(h^- (L_2)\). This leads both to a generalization and to a natural explanation of a formula appearing in the reviewer’s work [Ann. Acad. Sci. Fennicae, Ser. A I 416 (1967; Zbl 0153.37804)] about cyclotomic fields.

MSC:

11R21 Other number fields
11R29 Class numbers, class groups, discriminants
11R18 Cyclotomic extensions
11R27 Units and factorization
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