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Note on the class-number of the maximal real subfield of a cyclotomic field, II

Published online by Cambridge University Press:  22 January 2016

Hiroyuki Osada
Hiroyuki Osada*
Affiliation:
Department of Mathematics Rikkyo University, Ikebukuro, Tokyo 171, Japan
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For an integer m > 2, we denote by C(m) and H(m) the ideal class group and the class-number of the field

K = Q(ζm + ζm−1)

respectively, where ζm is a primitive m-th root of unity. Let q be a prime and /Q be a real cyclic extension of degree q. Let C() and h() be the ideal class group and the class-number of . In this paper, we give a relation between C() (resp. h()) and C(m) (resp. H(m)) in the case that m is the conductor of (Main Theorem). As applications of this main theorem, we give the following three propositions. In the previous paper [4], we showed that there exist infinitely many square-free integers m satisfying n|H(m) for any given natural number n. Using the result of Nakahara [2], we give first an effective sufficient condition for an integer m to satisfy n|H(m) for any given natural number n (Proposition 1). Using the result of Nakano [3], we show next that there exist infinitely many positive square-free integers m such that the ideal class group C(m) has a subgroup which is isomorphic to (Z/nZ)2 for any given natural number n (Proposition 2). In paper [4], we gave some sufficient conditions for an integer m to satisfy 3|H(m) and m≡l (mod 4). In this paper, using the result of Uchida [5], we give moreover a sufficient condition for an integer m to satisfy 4|H(m) and m ≡ 3 (mod 4) (Proposition 3). Finally, we give numerical examples of some square-free integers m satisfying 4|H(m) and m ≡ 3 (mod 4).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 113 , March 1989 , pp. 147 - 151
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1989

References

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[4] Osada, H., Note on the class-number of the maximal real subfield of a cyclotomic field, Manuscripta Math., 58 (1987), 215–227.CrossRefGoogle Scholar
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