Dihedral and cyclic extensions with large class numbers
Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 583-603.

This paper is a continuation of [2]. We construct unconditionally several families of number fields with large class numbers. They are number fields whose Galois closures have as the Galois groups, dihedral groups D n , n=3,4,5, and cyclic groups C n , n=4,5,6. We first construct families of number fields with small regulators, and by using the strong Artin conjecture and applying some modification of zero density result of Kowalski-Michel, we choose subfamilies such that the corresponding L-functions are zero free close to 1. For these subfamilies, the L-functions have the extremal value at s=1, and by the class number formula, we obtain large class numbers.

Cet article est la suite de [2]. Nous construisons inconditionnellement plusieurs familles de corps de nombres ayant un grand nombre de classes. Ce sont des corps de nombres dont la clôture galoisienne a pour groupe de Galois les groupes dièdraux D n , n=3,4,5, et les groupes cycliques C n , n=4,5,6. Nous construisons d’abord des familles de corps de nombres à petits régulateurs et, en utilisant la conjecture d’Artin forte et en appliquant une variante du résultat de densité nulle de Kowalski et Michel, nous choisissons des sous-familles telles que les fonctions L correspondantes soient sans zéro près de 1. Pour ces sous-familles, la fonction L prend une valeur extrémale en s=1 et, par la formule du nombre de classes, nous obtenons un grand nombre de classes.

DOI: 10.5802/jtnb.812
Peter J. Cho 1; Henry H. Kim 2

1 Department of Mathematics University of Toronto Toronto ON M5S 2E4 Canada
2 Department of Mathematics University of Toronto Toronto ON M5S 2E4 CANADA and Korea Institute for Advanced Study
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Peter J. Cho; Henry H. Kim. Dihedral and cyclic extensions with large class numbers. Journal de théorie des nombres de Bordeaux, Volume 24 (2012) no. 3, pp. 583-603. doi : 10.5802/jtnb.812. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.812/

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