On monogenity of certain pure number fields defined by $x^{p^r}-m$

Let $K = \mathbb{Q} (α)$ be a pure number field generated by a complex root $α$ 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 {$ν_p(m^p-m)=1$}, then $K$ is monogenic. But if $r\ge p$ and {$ν_p(m^{p}-m)> p$}, then $K$ is not monogenic. Some illustrating examples are given.

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