  Let $A$ be a finite-dimensional division algebra containing a base field $k$
in its center $F$. We say that $A$ is defined over a subfield $F_0$ of $F$ if
$A = A_0\otimes_{F_0} F$ for some $F_0$-subalgebra $A_0$ of $A$. We show that:
(1) In many cases $A$ can be defined over a rational extension of $k$. (2) If
$A$ has odd degree $n \ge 5$, then $A$ is defined over a field $F_0$ of
transcendence degree at most $(n-1)(n-2)/2$ over $k$. (3) If $A$ is a $Z/m
\times Z/2$-crossed product for some $m \ge 2$ (and in particular, if $A$ is
any algebra of degree 4) then $A$ is Brauer equivalent to a tensor product of
two symbol algebras. Consequently, $M_m(A)$ can be defined over a field $F_0$
of transcendence degree at most 4 over $k$. (4) If $A$ has degree 4 then the
trace form of $A$ can be defined over a field $F_0$ of transcendence degree at
most 4. (In (1), (3), and (4) we assume that the center of $A$ contains certain
roots of unity.)
