Fields of definition for division algebras

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 $(n-1)(n-2)\mathrm{/}2$ over $k$. (3) If $A$ is a $Z/m \times Z/2$-crossedproductforsome$m \ge 2$(andinparticular,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.)