# A refinement of B\'ezout's Lemma, and order 3 elements in some quaternion algebras over $\mathbb{Q}$

Abstract : Given coprime positive integers $d',d''$, B\'ezout'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 $|\alpha|^2$, where $\alpha\in\mathbb{Z}[j]$, the ring of integers of the number field $\mathbb{Q}(j)$, where $j^2+j+1=0$. We do this by using Atkin-Lehner elements in some quaternion algebras $\mathcal{H}$. We use this fact to count the number of conjugacy classes of elements of order 3 in an order $\mathcal{O}\subset\mathcal{H}$.
Document type :
Preprints, Working Papers, ...
Domain :

https://hal.archives-ouvertes.fr/hal-03285983
Contributor : Xavier Roulleau <>
Submitted on : Tuesday, July 13, 2021 - 5:23:53 PM
Last modification on : Wednesday, July 14, 2021 - 3:03:00 AM

### Identifiers

• HAL Id : hal-03285983, version 1
• ARXIV : 2107.02414

### Citation

Donald I. Cartwright, Xavier Roulleau. A refinement of B\'ezout's Lemma, and order 3 elements in some quaternion algebras over $\mathbb{Q}$. 2021. ⟨hal-03285983⟩

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