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The Dedekind-Hasse Criterion in Quaternion Algebras

This paper extends the Dedekind-Hasse criterion to quaternion orders to develop a finite algorithm for testing principal ideal domain status, which is then used to provide alternative proofs for specific non-Euclidean maximal orders and a completely arithmetic proof of Gordon Pall's result regarding norm-divisible elements and unique factorization.

Original authors: Adriana Cardoso, António Machiavelo

Published 2026-08-26

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Adriana Cardoso, António Machiavelo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast landscape of mathematics, there is a branch dedicated to understanding how numbers can be broken down into their most basic building blocks. For the familiar whole numbers we use every day, this process is straightforward and predictable: any number can be split into prime factors in only one way, much like a unique molecular structure. However, when mathematicians move beyond these simple numbers into more complex systems, this reliable rule often breaks down. In these intricate worlds, a single object might be able to be factored in multiple, conflicting ways, creating a chaotic environment where order is hard to find. To bring structure to this chaos, mathematicians look for special systems called principal ideal domains. These are rare, highly organized mathematical structures where the usual rules of factorization hold true, allowing for a clean and predictable breakdown of elements, just as we see with ordinary numbers.

The question of which complex number systems possess this special order has long been a puzzle, particularly for a specific family of objects known as quaternions. These are four-dimensional numbers that extend the concept of complex numbers, used in everything from 3D computer graphics to physics. Within the world of quaternions, there are special subsets called orders, which act like the integers within the broader system of all quaternions. For decades, mathematicians have known that some of these orders are perfectly organized, while others are not. A few specific cases, involving numbers with certain properties related to the number 7 and the number 13, remained stubbornly unsolved. While it was known that these specific systems were not "Euclidean"—a common, easy-to-check type of order that guarantees good behavior—it was not proven whether they still possessed the deeper, more subtle quality of being principal ideal domains.

A team of researchers from the University of Porto has now settled this question by developing a new, practical method to test these systems. They took a classic mathematical test, originally designed for simpler number systems, and adapted it to work within the complex, four-dimensional world of quaternions. This adaptation allowed them to create a finite, step-by-step procedure that could definitively determine if a given quaternion order is well-organized or not. Instead of relying on abstract theory alone, they turned this procedure into a computer algorithm. By feeding the specific cases of the orders with discriminant 7 and 13 into this algorithm, they were able to run a massive, exhaustive check. The computer examined millions of potential scenarios, looking for any sign of disorder.

The results were conclusive. The algorithm ran through every necessary check for the order associated with the number 7 in less than a second, finding no evidence of disorder. It then tackled the more complex case associated with the number 13, a task that required checking over 1.3 million specific configurations. This larger calculation took 45 minutes on a standard laptop, but it too returned a clean result. The researchers found that in both cases, the system behaved exactly as a principal ideal domain should. They proved that despite these systems lacking the simpler "Euclidean" property, they still possess the robust internal structure that allows for unique factorization. This means that even in these complex, non-Euclidean worlds, every element can still be broken down into prime components in a unique way, up to a specific type of rearrangement.

Beyond simply solving these two specific cases, the paper provides a powerful new tool for the field. The algorithm they created is not limited to just these two examples; it can be applied to any quaternion order to test its structural integrity. The researchers also used this method to provide a fresh, purely arithmetic proof of an older result regarding how elements in these systems can be divided. By showing that any element with a norm divisible by a certain integer must have a divisor with that exact norm, they reinforced the deep connection between the size of these numbers and their ability to be factored. The work confirms that the mathematical universe of quaternions is more orderly than previously assumed for these specific cases, and it offers a concrete, computational path for mathematicians to explore the structure of other complex number systems in the future.

Technical Summary: The Dedekind-Hasse's Criterion in Quaternion Algebras

Problem Statement
The paper addresses the problem of determining whether a specific order within a rational definite quaternion algebra is a Principal Ideal Domain (PID). While it is well-established that norm-Euclidean orders are PIDs, there exist maximal orders that are PIDs but are not norm-Euclidean. Specifically, the authors focus on the maximal orders of discriminant 7 and 13. The challenge lies in providing a constructive, arithmetic proof for these cases without relying on the theory of quadratic forms or non-constructive class number arguments.

Methodology
The authors develop a finite algorithm based on the Dedekind-Hasse criterion, adapted for non-commutative quaternion orders. The methodology proceeds in three main stages:

Theoretical Adaptation: The authors first establish that the Dedekind-Hasse criterion, originally formulated for commutative domains, is valid for quaternion orders. They prove that an order $H$ is a left PID if and only if for every $\rho \in A \setminus H$ , there exist $\alpha, \beta \in H$ such that $0 < N(\alpha\rho - \beta) < 1$ .
Reduction to a Finite Algorithm: To make the criterion computationally feasible, the authors reduce the infinite set of potential counterexamples ( $A \setminus H$ $A ∖ H$ ) to a finite set.
- They utilize the structure of the quotient space $A/H$ to restrict the search to elements with coefficients bounded by $1/2$ .
- They prove that it is sufficient to check elements of the form $\delta/p$ , where $p$ is a rational prime and $\delta \in H$ .
- Crucially, they derive an upper bound for the primes $p$ that need to be checked. Using a variation of Dirichlet's approximation theorem (Theorem 5) and bounding the norm of the remainder, they show that for sufficiently large $p$ , the condition is trivially satisfied. This reduces the verification to a finite number of primes and a finite set of quaternions for each prime.
- For the inner loop (finding $\alpha, \beta$ ), they employ Babai's nearest vector algorithm. By mapping the quaternion algebra to $\mathbb{R}^4$ via an isometry, the problem of finding $\beta$ becomes a Closest Vector Problem (CVP). The authors verify that the Hadamard ratio of the specific bases used is sufficiently close to 1, ensuring the algorithm's success in finding the required $\beta$ .
Arithmetic Proof of Pall's Result: Before applying the algorithm, the authors provide a purely arithmetic proof of a result by Gordon Pall. They demonstrate that in a PID order, if an element's norm is divisible by an integer $m$ , the element possesses a left and right divisor of norm $m$ . This establishes the existence and uniqueness (up to unit-migration) of factorizations modeled on the factorization of the norm.

Key Results
The paper applies the developed algorithm to two specific maximal orders:

Discriminant 7: The maximal order $H_{1,7}$ in the algebra $\left(\frac{-1, -7}{\mathbb{Q}}\right)$ . The algorithm required checking 102 quaternions across 6 primes ( $p < 163$ , refined to $p < 42$ ). The computation took less than one second.
Discriminant 13: The maximal order $H_{7,13}$ in the algebra $\left(\frac{-7, -13}{\mathbb{Q}}\right)$ . The algorithm required checking 1,339,411 quaternions across the first 30 primes ( $p < 393$ , refined to $p < 125$ ). The computation took 45 minutes on a standard laptop.

In both cases, the algorithm successfully found pairs $(\alpha, \beta)$ satisfying the Dedekind-Hasse condition for all required elements, thereby proving that these orders are PIDs.

Significance and Claims
The authors claim three primary contributions:

Alternative Proofs: They provide an alternative, arithmetic proof that the maximal orders of discriminant 7 and 13 are PIDs, despite not being norm-Euclidean. This complements previous work by Vigneras (class number 1) and Fitzgerald (basis construction).
Algorithmic Framework: They demonstrate that the Dedekind-Hasse criterion can be effectively transformed into a finite, executable algorithm for quaternion orders, bridging the gap between theoretical criteria and computational verification.
Arithmetic Factorization: They offer a completely arithmetic proof of Gordon Pall's theorem regarding divisors of specific norms in PID orders, avoiding the theory of quadratic forms. This reinforces the connection between norm factorization and element factorization in non-commutative settings.

The paper concludes by providing the PARI/GP code used to implement this algorithm, allowing for the verification of these results and the potential application to other orders.

Technical Summary: The Dedekind-Hasse's Criterion in Quaternion Algebras

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