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2012 Progress towards counting $D_5$ quintic fields
Eric Larson, Larry Rolen
Involve 5(1): 91-97 (2012). DOI: 10.2140/involve.2012.5.91


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)CX12 for some constant C. The best upper bound currently known is N(5,D5,X)X34+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 X23. 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.


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Eric Larson. Larry Rolen. "Progress towards counting $D_5$ quintic fields." Involve 5 (1) 91 - 97, 2012.


Received: 20 July 2011; Accepted: 4 August 2011; Published: 2012
First available in Project Euclid: 20 December 2017

zbMATH: 1279.11111
MathSciNet: MR2924317
Digital Object Identifier: 10.2140/involve.2012.5.91

Primary: 11R45
Secondary: 11R29 , 14G05

Keywords: Cohen–Lenstra heuristics for $p = 5$ , quintic dihedral number fields

Rights: Copyright © 2012 Mathematical Sciences Publishers


Vol.5 • No. 1 • 2012
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