On integral bases and monogenity of pure octic number fields with non-square free parameters

Abstract

In all available papers, on power integral bases of pure octic number fields $K$, generated by a root $\alpha$ of a monic irreducible polynomial $f(x)=x^8-m\in \mathbf{Z}[x]$, it was assumed that $m\ne \pm 1$ is square free. In this paper, we investigate the monogenity of any pure octic number field, without the condition that $m$ is square free. We start by calculating an integral basis of ${\mathbf{Z}}_K$, the ring of integers of $K$. In particular, we characterize when ${\mathbf{Z}}_K=\mathbf{Z}[\alpha ]$. We give sufficient conditions on $m$, which guarantee that $K$ is not monogenic. We finish the paper by investigating the case when $m=a^u$, $u\in \{1,3,5,7\}$ and $a\ne \mp 1$ is a square free rational integer.

arXiv admin note: text overlap with arXiv:2106.00004