An Embedding Property of Universal Division Algebras Original Research Article

Let A be a central simple algebra of degree n and let k be a subfield of its center. We show that A contains a copy of the universal division algebra Dm, n(k) generated by m generic n × n matrices if and only if trdegkA ≥ trdegkDm, n(k) = (m − 1)n2 + 1. Moreover, if in addition the center of A is finitely and separately generated over k then "almost all" division subalgebras of A generated by m elements are isomorphic to Dm, n(k). In the last section we give an application of our main result to the question of embedding free groups in division algebras.