Remarks on unit indices of imaginary abelian number fields
Let \(K\) be an imaginary abelian field. Within its unit group, the real units together with the roots of unity generate a subgroup of index \(Q_K=1\) or 2. This number is called the unit index of \(K\). A result proved by \textit{H. Hasse} in his monograph ``Über die Klassenzahl abelscher Zahlkörper'' [Berlin: Akademie-Verlag (1952; Zbl 0046.26003); reprint (1985; Zbl 0668.12003)] states that \(Q_K=1\) if \(f\), the conductor of \(K\), is a prime power. The present authors determine \(Q_ K\) in cases \(f\) is \(4p^a\), \(p^aq^b\) or \(2^np^a\) \((n\geq 3)\), where \(p\) and \(q\) are different odd primes and, in the third case, \(8\nmid p-1\). They also study \(Q_K\) for \(f=8p\) with \(8| p-1\), and for \(f=4pq\). The results provide many examples of cases in which \(Q_K=1\) but \(K\) contains a subfield \(k\) with \(Q_k=2\). It is pointed out how one should modify the places in Hasse's book [op. cit.] where it is erroneously assumed that \(Q_K\) be divisible by \(Q_k\).
counterexamples to Satz 29 in Hasse's monograph
scientific article; zbMATH DE number 4067039