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

Abstract

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 $m\ne 1$ 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.

Submitted. arXiv admin note: text overlap with arXiv:2006.11230