Prime factors of class number of cyclotomic fields

Tetsuya Taniguchi

doi:10.5802/jtnb.639

Tetsuya Taniguchi ¹

¹ Department of Mathematics, Tokyo University of Science, Noda, Chiba 278-8510, Japan

Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 525-530.

Abstract
Résumé

Let $p$ be an odd prime, $r$ be a primitive root modulo $p$ and $r_{i} \equiv r^{i} (mod p)$ with $1 \leq r_{i} \leq p - 1$ . In 2007, R. Queme raised the question whether the $ℓ$ -rank ( $ℓ$ an odd prime $\neq p$ ) of the ideal class group of the $p$ -th cyclotomic field is equal to the degree of the greatest common divisor over the finite field $𝔽_{ℓ}$ of $x^{(p - 1) / 2} + 1$ and Kummer’s polynomial $f (x) = \sum_{i = 0}^{p - 2} r_{- i} x^{i}$ . In this paper, we shall give the complete answer for this question enumerating a counter-example.

Soit $p$ un nombre premier impair, $r$ une racine primitive modulo $p$ et $r_{i} \equiv r^{i} (mod p)$ avec $1 \leq r_{i} \leq p - 1$ . En 2007, R. Queme a posé la question : le $ℓ$ -rang ( $ℓ$ premier impair $\neq p$ ) du groupe des classes d’idéaux du $p$ -ième corps cyclotomique est-il égal au degré du plus grand diviseur commun sur le corps fini $𝔽_{ℓ}$ de $x^{(p - 1) / 2} + 1$ et du polynôme de Kummer $f (x) = \sum_{i = 0}^{p - 2} r_{- i} x^{i}$ . Dans cet article, nous donnons une réponse complète à cette question en produisant un contre-exemple.

MR Zbl

DOI: 10.5802/jtnb.639

Author's affiliations:

Tetsuya Taniguchi ¹

¹ Department of Mathematics, Tokyo University of Science, Noda, Chiba 278-8510, Japan

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Tetsuya Taniguchi. Prime factors of class number of cyclotomic fields. Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, pp. 525-530. doi : 10.5802/jtnb.639. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.639/

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Cited by

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