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arXiv · 1906.11076

On quaternion algebras that split over quadratic number fields

Vincenzo Acciaro·Diana Savin·Mohammed Taous·Abdelkader Zekhnini
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Abstract

Let $d$ and $m$ be two distinct squarefree integers and $\mathcal{O}_K$ the ring of integers of the quadratic field $K=\mathbb{Q}(\sqrt{d})$. Denote by $ H_K(\alpha, m)$ a quaternion algebra over $K$, where $\alpha\in \mathcal{O}_K$. In this paper we give necessary and sufficient conditions for $ H_K(\alpha, m)$ to split over $K$ for some values of $\alpha$, and we obtain a complete characterization of division quaternion algebras $ H_K(\alpha, m)$ over $K$ whenever $\alpha$ and $m$ are two distinct positive prime integers. Examples are given involving prime Fibonacci numbers.

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Vincenzo Acciaro, Diana Savin, Mohammed Taous, Abdelkader Zekhnini. 2019-06-25. On quaternion algebras that split over quadratic number fields. https://arxiv.org/abs/1906.11076

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