Proceedings of the Japan Academy, Series A, Mathematical Sciences

Note on the ring of integers of a Kummer extension of prime degree. II

Humio Ichimura

Full-text: Open access

Abstract

Let $p$ be a prime number, and $a$ $(\in \mathbf{Q}^{\times})$ a rational number. Then, F. Kawamoto proved that the cyclic extension $\mathbf{Q}(\zeta_p, a^{1/p})/\mathbf{Q}(\zeta_p)$ has a normal integral basis if it is at most tamely ramified. We give some generalized version of this result replacing the base field $\mathbf{Q}$ with some real abelian fields of prime power conductor.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 2 (2001), 25-28.

Dates
First available in Project Euclid: 23 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148393122

Digital Object Identifier
doi:10.3792/pjaa.77.25

Mathematical Reviews number (MathSciNet)
MR1812042

Zentralblatt MATH identifier
0989.11062

Subjects
Primary: 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10]
Secondary: 11R18: Cyclotomic extensions

Keywords
Normal integral basis tame extension Kummer extension of prime degree

Citation

Ichimura, Humio. Note on the ring of integers of a Kummer extension of prime degree. II. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 2, 25--28. doi:10.3792/pjaa.77.25. https://projecteuclid.org/euclid.pja/1148393122


Export citation

References

  • Fröhlich, A.: Galois Module Structure of Algebraic Integers. Springer, Berlin-Heidelberg-New York (1983).
  • Gómez Ayala, E. J.: Bases normales d'entiers dans les extensions de Kummer de degré premier. J. Théor. Nombres Bordeaux, 6, 95–116 (1994).
  • Ichimura, H.: Note on the ring of integers of a Kummer extension of prime degree (2000) (preprint).
  • Iwasawa, K.: On some modules in the theory of cyclotomic fields. J. Math. Soc. Japan, 16, 42–82 (1964).
  • Kawamoto, F.: On normal integral bases. Tokyo J. Math., 7, 221–231 (1984).
  • Kawamoto, F.: Remark on “On normal integral bases”. Tokyo J. Math., 8, 275 (1985).
  • Kawamoto, F.: Normal integral bases and divisor polynomials, thesis, Gakushuin Univ. (1986).
  • van der Linden, F.: Class number calculation of real abelian number fields. Math. Comp., 39, 693–707 (1982).
  • Masley, J.: Class numbers of real cyclic number fields with small conductor. Compos. Math., 37, 297–319 (1978).
  • Mazur, B., and Wiles, A.: Class fields of abelian extensions over $\mathbf{Q}$. Invent. Math., 76, 179–330 (1984).
  • Washington, L.: Introduction to Cyclotomic Fields. 2nd ed., Springer, Berlin-Heidelberg-New York (1997).