Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Humio Ichimura

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Let $\ell$ be a prime number, and $K$ a number field with $\zeta_{\ell} \in K^{\times}$. We give a simple necessary and sufficient condition for all tame Kummer extensions over $K$ of degree $\ell$ to have a relative normal integral basis. The result is given in terms of the class number and the group of units of $K$.

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Proc. Japan Acad. Ser. A Math. Sci., Volume 78, Number 6 (2002), 76-79.

First available in Project Euclid: 23 May 2006

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

Normal integral basis Kummer extensions of prime degree


Ichimura, Humio. Note on the ring of integers of a Kummer extension of prime degree. V. Proc. Japan Acad. Ser. A Math. Sci. 78 (2002), no. 6, 76--79. doi:10.3792/pjaa.78.76.

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  • Fröhlich, A., and Taylor, M. J.: Algebraic Number Theory. Cambridge Univ. Press, Cambridge (1991).
  • Gómez Ayala, E. J.: Bases normales d'entiers dans les extensions de Kummer de degré premier. J. Théor. Nombres Bordeaux, 6, 95–116 (1994).
  • Greither, G., Replogle, D., Rubin, K., and Srivastav, A.: Swan modules and Hilbert-Speiser number fields. J. Number Theory, 79, 164–173 (1999).
  • Hasse, H.: Über die Klässenzahl abelscher Zahlkörper. Akademie-Verlag, Berlin (1952).
  • Ichimura, H.: Note on the ring of integers of a Kummer extension of prime degree, I (2000). (Preprint).
  • Ichimura, H.: On a normal integral basis problem over cyclotomic $\bZ_p$-extensions, II. Number Theory. (To appear).
  • Mann, H. B.: On integral basis. Proc. Amer. Math. Soc., 9, 167–172 (1958).
  • Uchida, K.: Imaginary abelian number fields of degree $2^m$ with class number one. Class Numbers and Fundamental Units of Algebraic Number Fields (eds. Yamamoto, Y., and Yokoi, H.). Proc. Internat. Conf., Katata, Japan, Nagoya Univ., Nagoya, pp. 156–170 (1986).
  • Washington, L. C.: Introduction to Cyclotomic Fields. 2nd ed., Springer, Berlin-Heidelberg-New York (1996).
  • Yamamura, K.: The determination of imaginary abelian number fields with class number one. Math. Comp., 62, 899–921 (1994).