On the monogenity of polynomials with non-squarefree discriminants

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 monogenic polynomials of degree $q$ with non-squarefree discriminants. In this article, using a method similar to the one 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, we present a class of non-monogenic polynomials whose coefficients are Stirling numbers of the first kind.