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The Dedekind-Hasse Criterion in Quaternion Algebras

Adriana Cardoso, António Machiavelo

TL;DR

This work extends the Dedekind-Hasse criterion from commutative domains to orders in quaternion algebras and shows how to turn it into a finite algorithm for deciding whether an order is a left (or right) PID. The authors apply the method to the maximal orders of discriminants $7$ and $13$, giving an explicit arithmetic proof that these non-Euclidean PIDs exist, alongside a purely arithmetic proof of Pall’s divisibility-factorization result. They connect factorization of quaternions to factorization of their norms and provide a PARI/GP implementation to realize the finite checks. The results offer new, non-Euclidean examples of PIDs in definite quaternion algebras and illuminate algorithmic pathways for identifying PID-quaternion orders in general.

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.

The Dedekind-Hasse Criterion in Quaternion Algebras

TL;DR

This work extends the Dedekind-Hasse criterion from commutative domains to orders in quaternion algebras and shows how to turn it into a finite algorithm for deciding whether an order is a left (or right) PID. The authors apply the method to the maximal orders of discriminants and , giving an explicit arithmetic proof that these non-Euclidean PIDs exist, alongside a purely arithmetic proof of Pall’s divisibility-factorization result. They connect factorization of quaternions to factorization of their norms and provide a PARI/GP implementation to realize the finite checks. The results offer new, non-Euclidean examples of PIDs in definite quaternion algebras and illuminate algorithmic pathways for identifying PID-quaternion orders in general.

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 always has a left and a right divisor with norm . This easily yields the existence and uniqueness (up to associates) of factorizations of a quaternion modeled on a factorization of its norm.

Paper Structure

This paper contains 8 sections, 14 theorems, 37 equations.

Table of Contents

  1. Introduction
  2. Factorization in a Principal Ideal Order
  3. The Dedekind-Hasse's Criterion
  4. Reduction to a Finite Algorithm
  5. The Maximal Orders of Discriminant 7 and 13
  6. Maximal Order of Discriminant 7
  7. Maximal Order of Discriminant 7
  8. PARI/GP Code

Key Result

Theorem 1

Let $H$ be a left (right) PID, $\theta \in H$ be a primitive quaternion, and $m \in \mathbb{N}$ be such that $m \mid \mathop{\rm N}\nolimits(\theta)$. Then there is only one right (left) divisor of $\theta$ of norm $m$, up to left (right) associates.

Theorems & Definitions (21)

  • Definition 1
  • Theorem 1
  • proof
  • Theorem 2: Factorization Theorem
  • Theorem 3
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • ...and 11 more