A NOTE ON THE MONOGENEITY OF POWER MAPS

Abstract

Let $\phi \left(x\right)=x^d-t\in \mathrm{\mathbb{Z}}\left[x\right]$ be an irreducible polynomial of degree $d\ge 2$, and let $\theta$ be a root of $\phi$. The purpose of this paper is to establish necessary and sufficient conditions for $\phi \left(x\right)$ to be monogenic, meaning the ring of integers of $\mathrm{\mathbb{Q}}\left(\theta \right)$ is generated by the powers of a root of $\phi \left(x\right)$. Sufficient conditions for monogeneity are established using Dedekind’s criterion. We then apply the Montes algorithm to give an explicit formula for the discriminant of $\mathrm{\mathbb{Q}}\left(\theta \right)$. Together, these results can be used to determine when $\phi \left(x\right)$ is not monogenic.