On quaternion algebra over the composite of quadratic number fields and over some dihedral fields

Let p and q be two positive primes. In this paper we obtain a complete characterization of quaternion division algebras H_K(p,q) over the composite K of n quadratic number fields. Also, in Section 6, we obtain a characterization of quaternion division algebras H_K(p,q) over some dihedral fields K.
This is a preliminary form of the article. Categories: number theory; rings and algebras