Discriminants of Fields Generated by Polynomials of Given Height

Abstract

We obtain upper bounds for the number of monic irreducible polynomials over $\mathbb{Z}$ 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).

Cited by in corpus (1)

• Quantitative Hilbert irreducibility and almost prime values of polynomial discriminants