On power integral bases for certain pure number fields

Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $f(x)=x^{12}-m$ with a square free rational integer $m\neq\mp 1$. 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. But if $m \equiv 1$ (mod $4$) or $m\equiv \mp 1$ (mod $9$), then the number field $K$ is not monogenic.