Proceedings of the Japan Academy, Series A, Mathematical Sciences

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

Humio Ichimura

Full-text: Open access

Abstract

Kawamoto [5, 6] proved that for any prime number $p$ and any $a \in \mathbf{Q}^{\times}$, the cyclic extenstion $\mathbf{Q}(\zeta_p, a^{1/p}) / \mathbf{Q}(\zeta_p)$ has a normal integral basis (NIB) if it is tame. We show that this property is peculier to the rationals $\mathbf{Q}$. Namely, we show that for a number field $K$ with $K \neq \mathbf{Q}$, there exist infinitely many pairs $(p, a)$ of a prime number $p$ and $a \in K^{\times}$ for which $K(\zeta_p, a^{1/p}) / K(\zeta_p)$ is tame but has no NIB. Our result is an analogue of the theorem of Greither et al. [3] on Hilbert-Speiser number fields.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 77, Number 6 (2001), 92-94.

Dates
First available in Project Euclid: 24 May 2006

Permanent link to this document
https://projecteuclid.org/euclid.pja/1148479942

Digital Object Identifier
doi:10.3792/pjaa.77.92

Mathematical Reviews number (MathSciNet)
MR1842864

Zentralblatt MATH identifier
0998.11065

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

Keywords
Normal integral basis Kummer extension of prime degree

Citation

Ichimura, Humio. Note on the ring of integers of a Kummer extension of prime degree. IV. Proc. Japan Acad. Ser. A Math. Sci. 77 (2001), no. 6, 92--94. doi:10.3792/pjaa.77.92. https://projecteuclid.org/euclid.pja/1148479942


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References

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