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

TL;DR

The paper investigates the monogenity of pure number fields $K = \mathbb{Q}(\sqrt[n]{m})$ for general $m$ and $n \ge 3$, without requiring $m$ to be square-free, by employing Ore's index theorem and Newton polygon techniques to study prime ideal decomposition and index forms. It establishes infinite families of non-monogenic fields via explicit conditions on $n= u p^r$ and $\nu_p(m^{p-1}-1)$, while also analyzing binomial cases $F(x)=x^n-a^u$ to delineate when $\mathbb{Z}_K$ equals $\mathbb{Z}[\alpha]$ or when a different primitive element yields a power integral basis, with CNS consequences. The results both extend prior work on monogenity and CNS for pure fields and provide constructive CNS bases in monogenic instances, highlighting the deep link between prime decomposition, residue degrees, and power integral bases. Overall, the work broadens the understanding of when pure number fields admit power integral bases and how this relates to the existence of canonical number systems, with explicit examples illustrating the methods and consequences. The methods offer a toolkit for determining CNS applicability in broader families of pure extensions beyond square-free settings.

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 Arnóczki 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}.

Figures (2)

• Figure 1: $N_{\phi_1}^+(F)$ with respect to $\nu_2$.
• Figure 2: $N_{\phi}^+(F)$ when $\nu \ge r+1$.

Theorems & Definitions (19)

• Theorem 2.1
• Remark 2.2
• Remark 2.3
• Corollary 2.4
• Corollary 2.5
• Corollary 2.6
• Corollary 2.7
• Theorem 2.8
• Remark 2.9
• Lemma 3.1