An embedding property of universal division algebras (Q1903026)

Let \(A\) be a central simple algebra of degree \(n\). The authors show that is contains a copy of the universal division algebra \(D_{m,n}(k)\) (i.e. the division algebra which is generated by \(m\) generic \(n\times n\) matrices over a field \(k\)) where \(k\) is a subfield of the center of \(A\) if and only if the degree of transcendence of \(A\) over \(k\) is at least the degree of transcendence of \(D_{m,n}(k)\). They show further that when it is the case then a ``general position'' division subalgebra of \(A\) which is generated by \(m\) elements is isomorphic to \(D_{m,n}(k)\). As an application of these results the authors give an ``elementary'' proof (i.e. a proof which does not depend on the Tits alternative) that a finite-dimensional division algebra with sufficiently large transcendence degree over its prime field contains a copy of a free non-commutative group.