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).

我们获得了\(\mathbb{Z}\)上固定次数n和生长高度H的一元不可约多项式的数量上限，其中由其根之一生成的场具有给定的判别式。我们通过两种互补方法对多项式判别式的无平方部分进行计数来实现此目的。反过来，这导致由次数为n且高度最多为H的多项式根生成的域的不同判别式的数量下界。我们还给出了具有给定判别式无平方部分的有界高度三项式数量的上限，改进了 IE Shparlinski (2010) 的先前结果。