Lower bounds for relative class numbers of CM-fields. (English) Zbl 0795.11058

Summary: Let \(K\) be a CM-field that is a quadratic extension of a totally real number field \(k\). Under a technical assumption, we show that the relative class number of \(K\) is large compared with the absolute value of the discriminant of \(K\), provided that the Dedekind zeta function of \(k\) has a real zero \(s\) such that \(0<s<1\). This result will enable us to get sharp upper bounds on conductors of totally imaginary abelian number fields with class number one or with prescribed ideal class groups.


11R29 Class numbers, class groups, discriminants
11R42 Zeta functions and \(L\)-functions of number fields
11R20 Other abelian and metabelian extensions
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