Remarks on unit indices of imaginary abelian number fields. II (Q915775)

This article continues the authors' study, begun in Part I [ibid. 60, 423--436 (1988; Zbl 0654.12002)], of the unit index \(Q_ K\) of an imaginary abelian field \(K\). Let \(h^*_ K\) denote the relative class number of \(K\). The authors determine \(Q_ K\) for some families of fields \(K\) of type \((2,2)\) or \((2,2,2)\), obtaining counter-examples to \textit{H. Hasse}'s Satz 41 [Über die Klassenzahl abelscher Zahlkörper. Berlin: Akademie-Verlag (1952; Zbl 0046.26003); reprints 1985; Zbl 0668.12003; Zbl 0668.12004)] about the quotient \(h^*_ K/h^*_ F\), where \(F\) is an imaginary subfield of \(K\). They propose a modified version for this Satz. (The incorrectness of the original formulation is due to a mistake in Satz 29 concerning the unit indices.) The authors also find a relation between the unit indices of certain fields \(K\) of type \((2,2^ 2)\) and their real cyclic subfields \(L\). The unit index of \(L\) is defined as the group index \([E_ L : H_ LE_ k]\), where \(E_ L\) and \(E_ k\) are the unit groups of \(L\) and its quadratic subfield, respectively, and \(H_ L=\{\eta \in E_ L:\) \(N_{L/k}(\eta)=\pm 1\}\).