# Dihedral and cyclic extensions with large class numbers

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

Journal de Théorie des Nombres de Bordeaux (2012)

- Volume: 24, Issue: 3, page 583-603
- ISSN: 1246-7405

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topCho, Peter J., and Kim, Henry H.. "Dihedral and cyclic extensions with large class numbers." Journal de Théorie des Nombres de Bordeaux 24.3 (2012): 583-603. <http://eudml.org/doc/251027>.

@article{Cho2012,

abstract = {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.},

affiliation = {Department of Mathematics University of Toronto Toronto ON M5S 2E4 Canada; Department of Mathematics University of Toronto Toronto ON M5S 2E4 CANADA and Korea Institute for Advanced Study},

author = {Cho, Peter J., Kim, Henry H.},

journal = {Journal de Théorie des Nombres de Bordeaux},

keywords = {class number; dihedral and cyclic extensions; L-functions},

language = {eng},

month = {11},

number = {3},

pages = {583-603},

publisher = {Société Arithmétique de Bordeaux},

title = {Dihedral and cyclic extensions with large class numbers},

url = {http://eudml.org/doc/251027},

volume = {24},

year = {2012},

}

TY - JOUR

AU - Cho, Peter J.

AU - Kim, Henry H.

TI - Dihedral and cyclic extensions with large class numbers

JO - Journal de Théorie des Nombres de Bordeaux

DA - 2012/11//

PB - Société Arithmétique de Bordeaux

VL - 24

IS - 3

SP - 583

EP - 603

AB - 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.

LA - eng

KW - class number; dihedral and cyclic extensions; L-functions

UR - http://eudml.org/doc/251027

ER -

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