On quaternion algebras that split over quadratic number fields

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.

arXiv admin note: substantial text overlap with arXiv:1802.08185