Monogenic polynomials with non-squarefree discriminant

We say a monic polynomial f(x)∈ Z[x] of degree n is monogenic if $f(x)$ is irreducible over Q and \1,θ ,θ ²,… ,θ ⁿ⁻¹ \ is a basis for the ring of integers of Q(θ ), where f(θ )=0. In 2012, for any integer n≥ 2, Kedlaya gave a construction to produce infinitely many monic integer-coefficient irreducible polynomials of degree n having squarefree discriminant. Such polynomials are necessarily monogenic. In this article, for any prime p≥ 3, we extend Kedlaya’s methods to construct explicit infinite families of monogenic polynomials of degree p having non-squarefree discriminant.