Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-27T03:16:54.205Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Galois module structure of ideal class groups

Published online by Cambridge University Press:  22 January 2016

Toru Komatsu  and
Shin Nakano
Toru Komatsu
Affiliation:
Department of Mathematics, Tokyo Metropolitan University, Hachioji, Tokyo 192-0397, Japan, trkomatu@comp.metro-u.ac.jp
Shin Nakano
Affiliation:
Department of Mathematics, Gakushuin University, Toshima-ku, Tokyo 171-8588, Japan, shin@math.gakushuin.ac.jp
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let K/k be a Galois extension of a number field of degree n and p a prime number which does not divide n. The study of the p-rank of the ideal class group of K by using those of intermediate fields of K/k has been made by Iwasawa, Masley et al., attaining the results obtained under respective constraining assumptions. In the present paper we shall show that we can remove these assumptions, and give more general results under a unified viewpoint. Finally, we shall add a remark on the class numbers of cyclic extensions of prime degree of Q.

Keywords

11R29
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 164 , December 2001 , pp. 133 - 146
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2001

References

[1] Cohen, H., Advanced topics in computational number theory, Springer-Verlag, New York, 2000.CrossRefGoogle Scholar
[2] Cornell, G., Group theory and the class group, Number theory and applications, NATO adv. Sci. Inst. Ser. C, 265 (1989), 347–352.Google Scholar
[3] Cornell, G. and Rosen, M., Group-theoretic constraints on the structure of the class group, J. Number Theory, 13 (1981), 1–11.CrossRefGoogle Scholar
[4] Dentzer, R.,, Polynomials with cyclic Galois group, Comm. in Algebra, 23 (1995), 1593–1603.CrossRefGoogle Scholar
[5] Iwasawa, K., A note on ideal class groups, Nagoya Math. J., 27 (1966), 239–247.CrossRefGoogle Scholar
[6] Lehmer, E., Connection between Gaussian period and cyclic units, Math Comp., 50 (1988), 535–541.CrossRefGoogle Scholar
[7] Masley, J. M., Class numbers of real cyclic number fields with small conductor, Com-positio Math., 37 (1978), 297–319.Google Scholar
[8] Serre, J-P., Topics in Galois theory, Jones and Bartlett Publishers, Boston, 1992.Google Scholar
[9] Washington, L. C., The non-p-part of the class number in a cyclotomic p-extension, Invent. Math., 49 (1978), 87–97.CrossRefGoogle Scholar
[10] Washington, L. C., Introduction to cyclotomic fields, 2nd edition, Springer-Verlag, New York, 1997.Google Scholar