Discriminants of fields generated by polynomials of given height

Abstract

We obtain upper bounds for the number of monic irreducible polynomials over $$\mathbb{Z}$$ Z of a fixed degree n and a growing height H for which the field generated by one of its roots has a given discriminant. We approach it via counting square-free parts of polynomial discriminants via two complementing approaches. In turn, this leads to a lower bound on the number of distinct discriminants of fields generated by roots of polynomials of degree n and height at most H. We also give an upper bound for the number of trinomials of bounded height with given square-free part of the discriminant, improving previous results of I. E. Shparlinski (2010).