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

TL;DR

This paper improves the upper bound on the number of degree n number fields with bounded discriminant for degrees between 6 and 94, using enhanced bounds on certain monic integer polynomials.

Contribution

It provides a tighter upper bound on the count of number fields with bounded discriminant, extending Schmidt's results through new polynomial bounds.

Findings

Improved upper bounds for number fields with degree 6 to 94.

Enhanced bounds on monic integer polynomials with specific discriminant divisibility.

Application of polynomial bounds to number field counting problem.

Abstract

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 $6 \leq n \leq 94$. 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.