In [+6] we proved Conjecture A if K is either an algebraic number field or the completion of an algebraic number field. The proofs given involved extremely technical number theoretic methods and were not applicable to other fields. In this paper we approach this question from a different viewpoint and prove a result valid for arbitrary fields. In the process we prove Conjecture A for a more general class of both K and D, and we obtain greatly simplified proofs of some of the main results of [4-61. By a K-division ring we mean a finite dimensional division algebra over K with center K. In view of Herstein’s result that finite subgroups of division rings of prime characteristic are cyclic [7], we restrict our attention to fields of characteristic zero. We let Q denote the field of rational numbers and, for G a finite multiplicative subgroup of D, we set 9(G) = {C a& ) ai E Q, Ai E G). g’(G) is a finite dimensional division algebra over Q (not necessarily central over Q). Amitsur in [2] determined the structure of g(G); in particular 93(G) depends on G up to isomorphism, and not on D. We denote the center of 93(G) by 8. Since D 3 g(G), D contains the subalgebra generated by K and g(G), which is easily seen to be KC? @ 99(G). We set A(G) = KC? OS S(G). We will maintain this notation throughout this paper.