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Remarks on unit indices of imaginary abelian number fields. (English) Zbl 0654.12002
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 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 \(Q_k\).

MSC:
11R20 Other abelian and metabelian extensions
11R27 Units and factorization
11R29 Class numbers, class groups, discriminants
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[2] Hasse, H.: über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (reproduction: Springer-Verlag, Berlin, 1985) · Zbl 0063.01966
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