Inclusion of CM-fields and divisibility of relative class numbers. (English) Zbl 0952.11023

Let \(K \subseteq L\) be CM-fields, i.e., totally complex quadratic extensions of totally real fields, and let \(h_K^-\) and \(h_L^-\) denote the minus class numbers of \(K\) and \(L\), respectively. In [Math. Z. 211, 505-521 (1992; Zbl 0761.11039)], K. Horie proved that \(h_K^- \mid 4h_L^-\) for abelian CM-fields, that is, for totally complex cyclotomic fields. In this paper, the author shows that this result is true for general (even non-normal) extensions of CM-fields. While Horie could use the analytic class number formula, the approach in the general case is via class field theory, and in fact the proof does not use analytic methods at all. In a first part, the result is shown to hold under various additional assumptions (for example, \(h_K^- \mid h_L^-\) if \((L:K)\) is odd); in the the second part, these results are used to show that any maximal subextension \(F/K\) with \(h_K^- \mid h_F^-\) satisfies \(h_F^- \mid 4h_L^-\), which implies the result.


11R29 Class numbers, class groups, discriminants
11R37 Class field theory


Zbl 0761.11039
Full Text: DOI EuDML