Proceedings of the Japan Academy, Series A, Mathematical Sciences

Normal integral basis and ray class group modulo 4

Humio Ichimura and Fuminori Kawamoto

Full-text: Open access

Abstract

We prove that a number field $K$ satisfies the following property (B) if and only if the ray class group of $K$ defined modulo 4 is trivial. (B): For any tame abelian extensions $N_1$ and $N_2$ over $K$ of exponent 2, the composite $N_1N_2/K$ has a relative normal integral basis (NIB) if both $N_1/K$ and $N_2/K$ have a NIB.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 79, Number 9 (2003), 139-141.

Dates
First available in Project Euclid: 18 May 2005

Permanent link to this document
https://projecteuclid.org/euclid.pja/1116443729

Digital Object Identifier
doi:10.3792/pjaa.79.139

Mathematical Reviews number (MathSciNet)
MR2022056

Zentralblatt MATH identifier
1059.11067

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

Keywords
Normal integral basis ray class group

Citation

Ichimura, Humio; Kawamoto, Fuminori. Normal integral basis and ray class group modulo 4. Proc. Japan Acad. Ser. A Math. Sci. 79 (2003), no. 9, 139--141. doi:10.3792/pjaa.79.139. https://projecteuclid.org/euclid.pja/1116443729


Export citation

References

  • Brinkhuis, J.: Unramified abelian extensions of CM-fields and their Galois module structure. Bull. London Math. Soc., 24, 236–242 (1992).
  • Childs, L.: The group of unramified Kummer extensions of prime degree. Proc. London Math. Soc., 35, 407–422 (1977).
  • Ichimura, H.: Note on the ring of integers of a Kummer extension of prime degree, V. Proc. Japan Acad., 78A, 76–79 (2002).
  • Kawamoto, F.: On quadratic subextensions of ray class fields of quadratic fields mod $\mathfrak{p}$. J. Number Theory, 86, 1–38 (2001).
  • Kawamoto, F.: Normal integral bases and strict ray class groups modulo 4. J. Number Theory, 101, 131–137 (2003).
  • Kawamoto, F., and Odai, Y.: Normal integral bases of $\infty$-ramified abelian extensions of totally real number fields. Abh. Math. Sem. Univ. Hamburg, 72, 217–233 (2002).
  • Massy, R.: Bases normales d'entiers relatives quadratiques. J. Number Theory, 38, 216–239 (1991).
  • Ono, T.: An Introduction to Algebraic Number Theory. Plenum Press, New York-London (1990).