On quaternion algebras that split over specific quadratic number fields

Let d and m be two distinct squarefree integers and (9K the ring of integers v/ of the quadratic field K = Q( d). Denote by HK(a, m) a quaternion algebra over K, where a E (9K. In this paper we give necessary and sufficient conditions for HK(a, m) to split over K for some values of a, and we obtain a complete characterization of division quaternion algebras HK(a, m) over K whenever a and m are two distinct positive prime integers. Examples are given involving prime Fibonacci numbers.