## Progress towards counting $$D_5$$ quintic fields.(English)Zbl 1279.11111

Summary: Let $$N(5,D_5,X)$$ 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 $$N(5,D_5,X) \sim C \cdot X^{\frac12}$$ for some constant $$C$$. The best upper bound currently known is $$N(5,D_5,X) \ll X^{\frac34+\varepsilon}$$, 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 $$\ll X^{\frac23}$$. Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of S. Wong [Proc. Am. Math. Soc. 133, No. 10, 2873–2881 (2005; Zbl 1106.11041)] on upper bounds for $$A_4$$ quartic fields in terms of a similar norm equation.

### MSC:

 11R45 Density theorems 11R29 Class numbers, class groups, discriminants

Zbl 1106.11041
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