Abstract.

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 D_m,n(k) generated by m generic n x n matrices if and only if trdeg_kA >= trdeg_kD_m,n(k) = (m-1)n²+1. Moreover, if in addition, the center of A is finitely and separably generated over k then  "almost all'' division  subalgebras of A generated by m elements are isomorphic to D_m,n(k).  In the last section we give an application of our main result to the question  of embedding free groups in division algebras.