On quaternion algebras that split over quadratic number fields

TL;DR

This paper characterizes when quaternion algebras over quadratic fields split or form division algebras, providing necessary and sufficient conditions, especially for cases involving prime integers and Fibonacci primes.

Contribution

It offers a complete characterization of division quaternion algebras over quadratic fields for specific prime parameters, extending understanding of their splitting behavior.

Findings

Necessary and sufficient conditions for splitting over quadratic fields.

Complete classification of division quaternion algebras with prime parameters.

Examples involving Fibonacci primes illustrate the theoretical results.

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.