Monogenic polynomials with non-squarefree discriminant

Abstract:

We say a monic polynomial $f(x)\in \mathbb {Z}[x]$ of degree $n$ is monogenic if $f(x)$ is irreducible over $\mathbb {Q}$ and $\left \{1,\theta ,\theta ^2,\ldots ,\theta ^{n-1}\right \}$ is a basis for the ring of integers of $\mathbb {Q}(\theta )$, where $f(\theta )=0$. In 2012, for any integer $n\ge 2$, 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 $p\ge 3$, we extend Kedlaya’s methods to construct explicit infinite families of monogenic polynomials of degree $p$ having non-squarefree discriminant.