Prime factors of class number of cyclotomic fields
Journal de théorie des nombres de Bordeaux, Volume 20 (2008) no. 2, p. 525-530

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

@article{JTNB_2008__20_2_525_0,
     author = {Taniguchi, Tetsuya},
     title = {Prime factors of class number of cyclotomic fields},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {20},
     number = {2},
     year = {2008},
     pages = {525-530},
     doi = {10.5802/jtnb.639},
     mrnumber = {2477516},
     zbl = {1163.11078},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2008__20_2_525_0}
}
Taniguchi, Tetsuya. 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. http://www.numdam.org/item/JTNB_2008__20_2_525_0/

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