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arXiv · 2506.22651

The Dedekind-Hasse Criterion in Quaternion Algebras

Adriana Cardoso·António Machiavelo
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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.

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BibTeXRIS

Adriana Cardoso, António Machiavelo. 2025-06-27. The Dedekind-Hasse Criterion in Quaternion Algebras. https://arxiv.org/abs/2506.22651

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