Squarefree discriminants of polynomials with prime coefficients

TL;DR

This paper investigates the frequency of polynomials with prime coefficients that have squarefree discriminants and maximal orders, contributing to understanding their algebraic properties.

Contribution

It provides the first comprehensive analysis of squarefree discriminants for polynomials with prime coefficients, both monic and non-monic.

Findings

Quantifies the number of such polynomials with squarefree discriminants

Establishes conditions for the maximality of the order in the quotient ring

Offers asymptotic estimates for the distribution of these polynomials

Abstract

In this paper, we consider the family of monic polynomials with prime coefficients and the family of all polynomials with prime coefficients. We determine the number of $f(x)$ in each of these families having: squarefree discriminant; $\mathbb{Z}[x]/(f(x))$ as the maximal order in $\mathbb{Q}[x]/(f(x))$.