Progress towards Counting D5 Quintic Fields

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 · X 1 2 for some constant C. The best known upper bound is N(5, D5, X) X 3 4, 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 X 23 . 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.