Nagoya Mathematical Journal

On the Galois module structure of ideal class groups

Toru Komatsu and Shin Nakano

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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 $\mathbb{Q}$.

Article information

Nagoya Math. J., Volume 164 (2001), 133-146.

First available in Project Euclid: 27 April 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11R29: Class numbers, class groups, discriminants


Komatsu, Toru; Nakano, Shin. On the Galois module structure of ideal class groups. Nagoya Math. J. 164 (2001), 133--146.

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