On the Monogenity of Polynomials with Non-Squarefree Discriminants

TL;DR

This paper constructs new infinite classes of monogenic polynomials with non-squarefree discriminants, extending previous work by Kedlaya and Jones, and explores polynomials with coefficients related to Stirling numbers.

Contribution

It introduces a novel class of monogenic polynomials with non-squarefree discriminants for primes of specific forms, expanding the understanding of polynomial monogenicity.

Findings

Constructed infinite classes of monogenic polynomials with non-squarefree discriminants.

Extended methods to primes of the form q = q0 + q1 - 1, with q0, q1 prime.

Presented non-monogenic polynomials with Stirling number coefficients.

Abstract

In 2012, for any integer $n \ge 2$, Kedlaya constructed an infinite class of monic irreducible polynomials of degree $n$ with integer coefficients having squarefree discriminants. Such polynomials are necessarily monogenic. Further, by extending Kedlaya's approach, for any odd prime $q$, Jones constructed a class of degree $q$ polynomials with non-squarefree discriminants. In this article, using a similar method provided by Jones, we present another infinite class of monogenic polynomials of degree $q$ with non-squarefree discriminants, where $q$ is a prime of the form $ q = q_0 + q_1 - 1 $, with $ q_0 $ and $ q_1 $ being prime numbers. In addition to this we present a class of non-monogenic polynomials whose coefficients are Sterling numbers of the first kind.