On power integral bases of certain pure number fields defined by $$x^{3^r} - m$$

Let $$K = \mathbb {Q} (\alpha )$$ be a pure number field generated by a root $$\alpha$$ of a monic irreducible polynomial $$F(x) = x^{3^r} -m$$ , with $$m \ne \pm 1$$ is a square-free rational integer and r is a positive integer. In this paper, we study the monogenity of K. We prove that if $$m \not \equiv \pm 1 \ \text { (mod }{9})$$ , then K is monogenic. We give also sufficient conditions on r and m for K to be not monogenic. Some illustrating examples are given too.