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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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