On the monogenity of pure number fields: application to the existence of canonical number systems

TL;DR

This paper investigates the monogenity of pure number fields generated by nth roots of integers, extending previous results to non-square-free cases and applying findings to the existence of canonical number systems.

Contribution

It generalizes the study of monogenity in pure number fields beyond square-free integers using Ore's theorem, and explores implications for canonical number systems.

Findings

Extended monogenity criteria to non-square-free integers

Provided examples illustrating the existence of canonical number systems

Connected monogenity properties with prime ideal decomposition

Abstract

Let $m$ be a rational integer with $m \neq 0, \pm 1$, and consider the pure number field $K = \mathbb{Q}(\sqrt[n]{m})$ with $n \ge 3$. Most papers discussing the monogenity of pure number fields focus exclusively on the case where $m$ is square-free. For every integer $n \ge 4$, the monogenity of number fields of degree $n$ is not completely characterized. For example, the monogenity of the pure quartic field $\mathbb{Q}(\sqrt[4]{m})$ is not yet fully described, even when $m$ is square-free (see the recent 2024 paper \cite{Nyul} by Arn\'oczki and Nyul). In this paper, based on a classical theorem of Ore concerning prime ideal decomposition in number fields \cite{MN92, O}, we study the monogenity of $K$ without assuming $m$ to be square-free. As an application, we present several examples related to canonical number systems (CNS). In particular, we observe that our results extend some of those presented in \cite{BFC, BF, HNHCNS}.