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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 fieldgenerated by one of its roots has a given discriminant. We approach it viacounting square-free parts of polynomial discriminants via two complementingapproaches. In turn, this leads to a lower bound on the number of distinctdiscriminants of fields generated by roots of polynomials of degree $n$ andheight at most $H$. We also give an upper bound for the number of trinomials ofbounded height with given square-free part of the discriminant, improvingprevious results of I. E. Shparlinski (2010).

Discriminants of Fields Generated by Polynomials of Given Height