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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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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
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