  Let $K = \mathbb{Q} (\alpha) $ be a pure number field generated by a complex
root $\alpha$ a monic irreducible polynomial $ F(x) = x^{p^r} -m$, with $ m
\neq 1 $ is a square free rational integer, $p$ is a rational prime integer,
and $r$ is a positive integer. In this paper, we study the monogenity of $K$.
We prove that if {{$\nu_p(m^p-m)=1$}}, then $K$ is monogenic. But if $r\ge p$
and {$\nu_p(m^{p}-m)> p$}, then $K$ is not monogenic. Some illustrating
examples are given.
