Hirabayashi, Mikihito; Yoshino, Ken-ichi Remarks on unit indices of imaginary abelian number fields. (English) Zbl 0654.12002 Manuscr. Math. 60, No. 4, 423-436 (1988). 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 \(Q_k\). Reviewer: Tauno Metsänkylä (Turku) Cited in 3 ReviewsCited in 10 Documents MSC: 11R20 Other abelian and metabelian extensions 11R27 Units and factorization 11R29 Class numbers, class groups, discriminants Keywords:class numbers; real units; unit index; counterexamples to Satz 29 in Hasse’s monograph Citations:Zbl 0046.26003; Zbl 0668.12003 PDFBibTeX XMLCite \textit{M. Hirabayashi} and \textit{K.-i. Yoshino}, Manuscr. Math. 60, No. 4, 423--436 (1988; Zbl 0654.12002) Full Text: DOI EuDML References: [1] Hasse, H.: Bericht über neuere Untersuchungen und Probleme aus der Theorie der algebraischen Zahlkörper, Teil I, la und II, Physica-Verlag, Würzburg-Wien, 1965 · Zbl 0138.03202 [2] Hasse, H.: über die Klassenzahl abelscher Zahlkörper, Akademie-Verlag, Berlin, 1952 (reproduction: Springer-Verlag, Berlin, 1985) · Zbl 0063.01966 [3] Hirabayashi, M. and Yoshino, K.: The Unit Indices of Imaginary Abelian Number Fields, Proc. Japan Acad.60 Ser. A, 215-217 (1984) · Zbl 0544.12002 · doi:10.3792/pjaa.60.215 [4] Hirabayashi, M. and Yoshino, K.: On the Relative Class Number of the Imaginary Abelian Number Field I and II, Memoirs of the College of Liberal Arts and Kanazawa Medical University, vol.9, 5-53 (1981) and vol.10, 33-81 (1982) [5] Iwasawa, K.: A note on class numbers of algebraic number fields, Abh. Math. Sem. Univ. Hamburg20, 257-258 (1956) · Zbl 0074.03002 [6] Washington, L. C.: Introduction to Cyclotomic Fields, Springer-Verlag, New York, 1982 · Zbl 0484.12001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.