Weber’s class number problem in the cyclotomic 2 -extension of , II
Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 359-368.

Let h n denote the class number of n-th layer of the cyclotomic 2 -extension of . Weber proved that h n (n1) is odd and Horie proved that h n (n1) is not divisible by a prime number satisfying 3,5(mod8). In a previous paper, the authors showed that h n (n1) is not divisible by a prime number less than 10 7 . In this paper, by investigating properties of a special unit more precisely, we show that h n (n1) is not divisible by a prime number less than 1.2·10 8 . Our argument also leads to the conclusion that h n (n1) is not divisible by a prime number satisfying ¬±1(mod16).

Soit h n le nombres de classes du n-ième étage de la 2 -extension cyclotomique de . Weber a prouvé que h n (n1) est impair et Horie a prouvé que h n (n1) n’est divisible par aucun nombre premier satisfaisant 3,5(mod8). Dans un article précédent, les auteurs ont montré h n (n1) n’est divisible par aucun nombre premier inférieur à 10 7 . Dans le présent article, en étudiant plus précisément les propriétés d’une unité particulière, nous montrons que h n (n1) n’est divisible par aucun nombre premier inférieur à 1.2·10 8 . Notre argument conduit aussi à la conclusion que h n (n1) n’est divisible par aucun nombre premier satisfaisant ¬±1(mod16).

DOI: 10.5802/jtnb.720
Fukuda, Takashi 1; Komatsu, Keiichi 2

1 Department of Mathematics College of Industrial Technology Nihon University 2-11-1 Shin-ei, Narashino, Chiba, Japan
2 Department of Mathematics School of Science and Engineering Waseda University 3-4-1 Okubo, Shinjuku, Tokyo 169-8555, Japan
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Fukuda, Takashi; Komatsu, Keiichi. Weber’s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, II. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 2, pp. 359-368. doi : 10.5802/jtnb.720. http://archive.numdam.org/articles/10.5802/jtnb.720/

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