Proceedings of the Japan Academy, Series A, Mathematical Sciences

Note on imaginary quadratic fields satisfying the Hilbert-Speiser condition at a prime p

Humio Ichimura

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Abstract

Let $p$ be a prime number. A number field $F$ satisfies the condition $(H_p)$ when any tame cyclic extention $N/F$ of degree $p$ has a normal integral basis. For the case $p=2$, it is shown by Mann that $F$ satisfies $(H_2)$ only when $h_F=1$ where $h_F$ is the class number of $F$. We prove that if an imaginary quadratic field $F$ satisfies $(H_p)$ for some $p$, then $h_F=1$.

Article information

Source
Proc. Japan Acad. Ser. A Math. Sci., Volume 83, Number 6 (2007), 88-91.

Dates
First available in Project Euclid: 29 August 2007

Permanent link to this document
https://projecteuclid.org/euclid.pja/1188405577

Digital Object Identifier
doi:10.3792/pjaa.83.88

Mathematical Reviews number (MathSciNet)
MR2355504

Zentralblatt MATH identifier
1157.11044

Subjects
Primary: 11R33: Integral representations related to algebraic numbers; Galois module structure of rings of integers [See also 20C10] 11R11: Quadratic extensions

Keywords
Hilbert-Speiser number field imaginary quadratic field

Citation

Ichimura, Humio. Note on imaginary quadratic fields satisfying the Hilbert-Speiser condition at a prime p. Proc. Japan Acad. Ser. A Math. Sci. 83 (2007), no. 6, 88--91. doi:10.3792/pjaa.83.88. https://projecteuclid.org/euclid.pja/1188405577


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