Abstract: This is a significantly revised and expanded version of the earlier preprint ``Lattices and parameter reduction in division algebras", written jointly with M. Lorenz (see below). 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 if there exists an F_0-algebra A_0 such that A = A_0 \otimes_{F_0} F. We ask when F_0 can be chosen to be rational over k or have low transcendence degree over k. We show that: (i) In many cases A can be defined over a rational extension of k. (ii) If A has odd degree n >= 5, then A is defined over a field F_0 of transcendence degree <= 1/2 (n-1)(n-2) over k. (iii) A is an algebra of degree 4 then A is Brauer equivalent to a tensor product of two symbol algebras. Consequently, M_2(A) can be defined over a field F_0 such that trdeg_k(F_0) <= 4. (iv) If A has degree 4 then the trace form of A can be defined over a field F_0 of transcendence degree 4. (In (i), (iii) and (iv) we assume that the center of A contains certain roots of unity.) Our results in (iii) and (iv) complement a recent theorem of Rost, which asserts that a generic degree 4 algebra cannot be defined over a field F_0 with <= 4.