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

TL;DR

This paper studies the monogenity of pure octic number fields generated by roots of x^8 - m, removing the usual assumption that m is square free, and provides criteria for when these fields are monogenic or not.

Contribution

It extends the analysis of pure octic fields by removing the square-free restriction on m and characterizes when the ring of integers equals the generated order.

Findings

Calculated integral bases for various pure octic fields.

Provided conditions under which these fields are not monogenic.

Analyzed special cases where m is a power of a square-free integer.

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 $m\neq \pm 1$ 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 $a\neq \mp 1$ is a square free rational integer.