  We define a transcendence degree for division algebras, by modifying the
lower transcendence degree construction of Zhang. We show that this invariant
has many of the desirable properties one would expect a noncommutative analogue
of the ordinary transcendence degree for fields to have. Using this invariant,
we prove the following conjecture of Small. Let $k$ be a field, let $A$ be a
finitely generated $k$-algebra that is an Ore domain, and let $D$ denote the
quotient division algebra of $A$. If $A$ does not satisfy a polynomial identity
then the Gelfand-Kirillov dimension of $K$ is at most the Gelfand-Kirillov
dimension of $A$ minus 1 for every commutative subalgebra $K$ of $D$.
