  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).
