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November 2011 A note on the divisibility of class numbers of imaginary quadratic fields $\mathbf{Q}(\sqrt{a^{2} - k^{n}})$
Akiko Ito
Proc. Japan Acad. Ser. A Math. Sci. 87(9): 151-155 (November 2011). DOI: 10.3792/pjaa.87.151

Abstract

Let $n > 1$ be an integer, $k > 1$ be an odd integer and $a > 0$ be an even integer. Suppose $a^{2} + b^{2}d = k^{n}$, where $d \neq 1, 3$ is a positive odd square-free integer and $\gcd(a, bd) = 1$. In this paper, we describe imaginary quadratic fields $\mathbf{Q}(\sqrt{a^{2} - k^{n}})$ explicitly whose class numbers are divisible by $n$ if $d \equiv 1, 5, 7\bmod 8$ or $d \equiv 3\bmod 8$ with $(n, 3) = 1$ under certain conditions.

Citation

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Akiko Ito. "A note on the divisibility of class numbers of imaginary quadratic fields $\mathbf{Q}(\sqrt{a^{2} - k^{n}})$." Proc. Japan Acad. Ser. A Math. Sci. 87 (9) 151 - 155, November 2011. https://doi.org/10.3792/pjaa.87.151

Information

Published: November 2011
First available in Project Euclid: 4 November 2011

zbMATH: 1247.11139
MathSciNet: MR2863357
Digital Object Identifier: 10.3792/pjaa.87.151

Subjects:
Primary: 11D61 , 11R11 , 11R29

Keywords: class numbers , imaginary quadratic fields , primitive divisors of Lucas numbers

Rights: Copyright © 2011 The Japan Academy

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Vol.87 • No. 9 • November 2011
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