On power integral bases of certain pure number fields defined by $x^{42} - m$

Abstract

Let $K = {\mathbb {Q}} (\alpha )$ be a pure number field generated by a complex root $\alpha$ of a monic irreducible polynomial $F(x) = x^{42} -m \in {{\mathbb {Z}}}[x]$, where $m \ne \pm 1$ is a square-free rational integer. In this paper, we study the monogenity of K. We prove that if $m\not \equiv 1\ \mathrm{(mod }{4})$, $m\not \equiv \mp 1 \ \mathrm{(mod }{9})$, and $\overline{m}\not \in \{\mp 1, 18, 19, 30, 31\} \ \mathrm{(mod }{49})$, then K is monogenic. But, if $m \equiv 1\ \mathrm{(mod }{4})$, or $m \equiv 1 \ \mathrm{(mod }{9})$, or $m \equiv 1 \ \mathrm{(mod }{49})$, then K is not monogenic. Our results are illustrated by some examples.