On power integral bases of certain pure number fields defined by $X^{60}-m$

TL;DR

This paper investigates the monogeneity of pure number fields generated by roots of the polynomial x^{60} - m, establishing specific congruence conditions under which these fields are monogenic or not.

Contribution

It provides a complete characterization of monogeneity for these fields based on congruence conditions of m, extending understanding of power integral bases in pure number fields.

Findings

Fields are monogenic if m does not satisfy certain congruences.

Fields are not monogenic if m satisfies specific congruence conditions.

Examples illustrate the theoretical results.

Abstract

Let $K$ be a pure number field generated by a complex root of a monic irreducible polynomial $F(x)=x^{60}-m\in \mathbb{Z}[x]$, with $m\neq \pm1$ a square free integer. In this paper, we study the monogeneity of $K$. We prove that if $m\not\equiv 1\md{4}$, $m\not\equiv \mp 1 \md{9} $ and $\overline{m}\not\in\{\mp 1,\mp 7\} \md{25}$, then $K$ is monogenic. But if $m\equiv 1\md{4}$, $m\equiv \mp1 \md{9}$, or $m\equiv \mp 1\md{25}$, then $K$ is not monogenic. Our results are illustrated by examples.