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arXiv · 1909.00135

Discriminants of Fields Generated by Polynomials of Given Height

Rainer Dietmann·Alina Ostafe·Igor E. Shparlinski
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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).

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Rainer Dietmann, Alina Ostafe, Igor E. Shparlinski. 2019-08-31. Discriminants of Fields Generated by Polynomials of Given Height. https://arxiv.org/abs/1909.00135

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