On Power integral bases for certain pure number fields

Let $K=\mathbb{Q}(\alpha)$ be a number field generated by a complex root $\alpha$ of a monic irreducible polynomial $f(x)=x^{12}-m$, with $m\neq 1$ is a square free rational integer. In this paper, we prove that if $m \equiv 2$ or $3$ (mod 4) and $m\not\equiv \mp 1$ (mod 9), then the number field $K$ is monogenic. If $m \equiv 1$ (mod 8) or $m\equiv \mp 1$ (mod 9), then the number field $K$ is not monogenic.