Let be the number of quintic number fields whose Galois closure has Galois group and whose discriminant is bounded by . By a conjecture of Malle, we expect that for some constant . The best upper bound currently known is , 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 . Finally, we show how such norm equations can be helpful by reinterpreting an earlier proof of Wong on upper bounds for quartic fields in terms of a similar norm equation.
"Progress towards counting $D_5$ quintic fields." Involve 5 (1) 91 - 97, 2012. https://doi.org/10.2140/involve.2012.5.91