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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

Citations:

Zbl 1106.11041
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