Counting cyclic quartic extensions of a number field
type: Item
instance of: scholarly article
title: Counting cyclic quartic extensions of a number field
zbMATH Open document ID: 1090.11068
author: Henri Cohen
author: Francisco Diaz y Diaz
author: Michel Olivier
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2006-03-30T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/52430
review text: Let \(G\) be a finite group and \(K\) some number field. Let \(N_K(G,X)\) denote the number of all Galois extensions \(L/K\) with Galois group \(G\) such that the norm of the relative discriminant is \(\leq X\). By a conjecture of Malle it is expected that there are constants \(a_K, b_K, c_K\) (depending on \(G\)) such that \[ N_K(G,X) = c_K X^{a_K} (\log X)^{b_K-1}. \] The constants \(a_K\) and \(b_K\) were determined by \textit{D. J. Wright} [Proc. Lond. Math. Soc. (3) 58, No. 1, 17--50 (1989; Zbl 0628.12006)] for all abelian groups; in this article, the authors determine \(c_K\) in the case where \(G\) is the cyclic group of order \(4\).
reviewed by: Franz Lemmermeyer
Mathematics Subject Classification ID: 11R16
Mathematics Subject Classification ID: 11R45
Mathematics Subject Classification ID: 11R29
Mathematics Subject Classification ID: 11R32
zbMATH DE Number: 5016570
zbMATH Keywords: density
zbMATH Keywords: discriminants
zbMATH Keywords: cyclic quartic extensions
zbMATH Keywords: ramification
MaRDI profile type: Publication
OpenAlex ID: W2011998986
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DOI: 10.5802/JTNB.503
description: scientific article; zbMATH DE number 5016570