A refinement of Bézout's Lemma, and order 3 elements in some quaternion algebras over ℚ

2021/07/06 by Donald I. Cartwright, Cartwright, Donald I., Xavier Roulleau +1 · 1 citation
Mathematics · #11R52 #Algebraic Geometry and Number Theory #FOS: Mathematics #Mathematical Dynamics and Fractals #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2107.02414

openalex publication_date 2021/07/06 · openalex created_date 2021/07/19 · openalex updated_date 2026/07/28

Abstract

Given coprime positive integers d',d'', Bézout's Lemma tells us that there are integers u,v so that d'u-d''v=1. We show that, interchanging d' and d'' if necessary, we may choose u and v to be Loeschian numbers, i.e., of the form |α|², where α∈ℤ[j], the ring of integers of the number field ℚ(j), where j²+j+1=0. We do this by using Atkin-Lehner elements in some quaternion algebras H. We use this fact to count the number of conjugacy classes of elements of order 3 in an order O\subsetH.

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