Locally Associated Orders in Real Quadratic Number Fields

TL;DR

This paper investigates locally associated orders in real quadratic number fields, providing new methods for their classification and proposing a conjecture related to Pell's equations to explain exceptional cases.

Contribution

It introduces strategies to determine local association of orders in real quadratic fields and formulates a conjecture on Pell's equations for unresolved cases.

Findings

Simplified criteria for identifying locally associated orders.

Identification of exceptional cases that resist classification.

A conjecture linking Pell's equations to these exceptional cases.

Abstract

In 2025, the concept of an order in a number field being associated, ideal-preserving, or locally associated was introduced in order to tackle problems in factorization. In this paper, we explore locally associated orders in real quadratic number fields of the form $\mathbb{Q}[\sqrt{p}]$, with $p\in\mathbb{N}$ prime. In particular, we develop strategies and produce results which make determining when a given order in such a number field is (or is not) locally associated much easier. We also highlight the relatively few cases which defy simple characterization, leading to a conjecture on the solutions to Pell's equations of the form $x^2-y^2p=1$.