On quaternion algebras over some extensions of quadratic number fields

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