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 $N\left(5,{D}_{5},X\right)$ be the number of quintic number fields whose Galois closure has Galois group ${D}_{5}$ and whose discriminant is bounded by $X$. By a conjecture of Malle, we expect that $N\left(5,{D}_{5},X\right)\sim C\cdot {X}^{\frac{1}{2}}$ for some constant $C$. The best upper bound currently known is $N\left(5,{D}_{5},X\right)\ll {X}^{\frac{3}{4}+\epsilon }$, 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 $\ll {X}^{\frac{2}{3}}$. Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for ${A}_{4}$ 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