On quaternion algebras over some extensions of quadratic number fields

Let p and q be two positive primes, let l be an odd positive prime and let F be a quadratic number field. Let K be an extension of F of degree l such that K is a dihedral extension of Q, or else let K be an abelian l-extension of F unramified over F whenever l divides the class number of F. In this paper, we provide a complete characterization of division quaternion algebras H-K(p, q) over K.