Prime factors of class number of cyclotomic fields

Taniguchi, Tetsuya

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

Let $p$ be an odd prime, $r$ be a primitive root modulo $p$ and $r_i\equiv r^i(modp)$ with $1\le r_i\le p-1$. In 2007, R. Queme raised the question whether the $\ell$-rank ($\ell$ an odd prime $\ne p$) 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 ${𝔽}_{\ell }$ of $x^{(p-1)/2}+1$ and Kummer’s polynomial $f\left(x\right)={\sum }_{i=0}^{p-2}{r}_{-i}{x}^i$. 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 $r_i\equiv r^i(modp)$ avec $1\le r_i\le p-1$. En 2007, R. Queme a posé la question : le $\ell$-rang ($\ell$ premier impair $\ne p$) 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 ${𝔽}_{\ell }$ de $x^{(p-1)/2}+1$ et du polynôme de Kummer $f\left(x\right)={\sum }_{i=0}^{p-2}{r}_{-i}{x}^i$. Dans cet article, nous donnons une réponse complète à cette question en produisant un contre-exemple.

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