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Monogenesis of the rings of integers in certain imaginary abelian fields

Published online by Cambridge University Press:  22 January 2016

Syed Inayat Ali Shah  and
Toru Nakahara
Syed Inayat Ali Shah
Affiliation:
Department of Engineering Systems and Technology, Course of Science and Engineering, Graduate School of Saga University, Saga 840-8502, Japan, shah@ms.saga-u.ac.jp
Toru Nakahara
Affiliation:
Department of Mathematics, Faculty of Science and Engineering, Saga University, Saga 840-8502, Japan, nakahara@ms.saga-u.ac.jp
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Abstract

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In this paper we consider a subfield K in a cyclotomic field km of conductor m such that [km: K] = 2 in the cases of m = lpn with a prime p, where l = 4 or p > l = 3. Then the theme is to know whether the ring of integers in K has a power basis or does not.

Keywords

11R1811R04

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 168 , 2002 , pp. 85 - 92
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2002

References

[DK] Dummit, D. S. and Kisilevsky, H., Indices in cyclic cubic fields, Number Theory and Algebra, Collect. Pap. Dedic. H. B. Mann, Ross, A. E. and Taussky-Todd, O., New York San Francisco London, Academic Press, 1977, 29–42.Google Scholar
[Ga] Gaál, I., Computing all power integral bases in orders of totally real cyclic sextic number fields, Math. Comp., 65 (1996), 801–822.Google Scholar
[Gr] Gras, M.-N., Non monogénéité de l’anneau des extensions cycliques de de degré premier l ≥ 5, J. Number Theory, 23 (1986), 347–353.Google Scholar
[HSW] Huard, J. G., Spearman, B. K. and Williams, K. S., Integral Bases for Quartic Fields with Quadratic Subfields, J. Number Theory, 51 (1995), 87–102.Google Scholar
[Li] Liang, J, On integral basis of the maximal real subfield of a cyclotomic field, J. Reine Angew. Math., 286/287 (1976), 223–226.Google Scholar
[N1] Nakahara, T., On cyclic biquadratic fields related to a problem of Hasse, Mh. Math., 94 (1982), 125–132.Google Scholar
[N2] Nakahara, T., A simple proof for non-mono genesis of the rings of integers in some cyclic Fields, the Proceedings of the third Conference of the Canadian Number Theory Association, Oxford, Clarendon Press, 1993, 167–173.Google Scholar
[Nar] Narkiewicz, W., Elementary and Analytic Theory of Algebraic Numbers, 2nd Edition, Springer-Verlag; Warszawa, PWN-Polish Scientific Publishers, Berlin Heidelberg New York, 1990.Google Scholar
[SN] Shah, S. I. A., and Nakahara, T., Non-monogenetic aspect of the rings of integers in certain abelian fields, the Proceedings of the Jangjeon Mathematical Society, Pusan, Ku-Deok Co., 1, 2000, 75–79.Google Scholar
[T] Thérond, J.-D., Existence d’une extension cyclique monogene de discriminant donné, Arch. Math., 41 (1983), 243–255.Google Scholar
[W] Washington, C. L., Introduction to cyclotomic fields, 2nd Edition GTM 83 1997, Springer-Verlag., New York Heidelberg Berlin.Google Scholar