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

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).

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).

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DOI : https://doi.org/10.5802/jtnb.720
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     author = {Takashi Fukuda and Keiichi Komatsu},
     title = {Weber{\textquoteright}s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, {II}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {359--368},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
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}
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Takashi Fukuda; Keiichi Komatsu. Weber’s class number problem in the cyclotomic $\mathbb{Z}_2$-extension of $\mathbb{Q}$, II. Journal de Théorie des Nombres de Bordeaux, Tome 22 (2010) no. 2, pp. 359-368. doi : 10.5802/jtnb.720. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.720/

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