The Dedekind-Hasse Criterion in Quaternion Algebras

TL;DR

This paper extends Dedekind-Hasse's PID criterion to quaternion orders, providing an algorithm to identify principal ideal domains and offering new proofs for properties of maximal orders and element factorizations in quaternion algebras.

Contribution

It adapts a classical PID criterion to quaternion orders, introduces an algorithm for identifying principal ideal domains, and offers new proofs for properties of maximal orders and element factorizations.

Findings

An algorithm to determine if a quaternion order is a PID.

Maximal orders of discriminant 7 and 13 are PIDs.

Existence and uniqueness of factorizations in PIDs.

Abstract

We show that a criterion for an integral domain to be a principal ideal domain (PID), due to Dedekind and Hasse, can also be applied in quaternion orders, and that it can be used to build a finite algorithm to determine if a given order is a principal left (or right) ideal domain. Using this algorithm, we give an alternative proof that the maximal orders of discriminant 7 and 13, which are non-Euclidean, are PIDs. We also provide a completely arithmetic proof of a result of Gordon Pall that shows that, in an order that is a PID, an element of whose norm is divisible by an integer $m$ always has a left and a right divisor with norm $m$. This easily yields the existence and uniqueness (up to associates) of factorizations of a quaternion modeled on a factorization of its norm.