Vol. 5, No. 1, 2012

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Progress towards counting $D_5$ quintic fields

Eric Larson and Larry Rolen

Vol. 5 (2012), No. 1, 91–97
Abstract

Let N(5,D5,X) be the number of quintic number fields whose Galois closure has Galois group D5 and whose discriminant is bounded by X. By a conjecture of Malle, we expect that N(5,D5,X) C X1 2 for some constant C. The best upper bound currently known is N(5,D5,X) X3 4 +ε, and we show this could be improved by counting points on a certain variety defined by a norm equation; computer calculations give strong evidence that this number is X2 3 . Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for A4 quartic fields in terms of a similar norm equation.

Keywords
quintic dihedral number fields, Cohen–Lenstra heuristics for $p = 5$
Mathematical Subject Classification 2010
Primary: 11R45
Secondary: 11R29, 14G05
Milestones
Received: 20 July 2011
Accepted: 4 August 2011
Published: 28 April 2012

Communicated by Ken Ono
Authors
Eric Larson
Department of Mathematics
Harvard
Cambridge, MA 02138
United States
Larry Rolen
Department of Mathematics and Computer Science
Emory University
Atlanta, GA 30322
United States