Explicit integral basis of pure sextic fields

Abstract

Let $K=\mathbb{Q}\left(𝜃\right)$ be an algebraic number field with $𝜃$ satisfying an irreducible polynomial $x^6-m$ having integer coefficient $m$. We explicitly provide an integral basis of $K$ along with some examples. As an application, we show that when $m$ is a squarefree integer such that $m\equiv 2,3mod4$ and $m\not\equiv \pm 1mod9$, then $K$ has a power integral basis.