An improvement on Schmidt’s bound on the number of number fields of bounded discriminant and small degree

We prove an improvement on Schmidt’s upper bound on the number of number fields of degree n and absolute discriminant less than X for 6≤ n≤ 94 . We carry this out by improving and applying a uniform bound on the number of monic integer polynomials, having bounded height and discriminant divisible by a large square, that we proved in a previous work [7].