Fields of definition for division algebras

Let A be a finite-dimensional division algebra containing a base field k in its center F. A is defined over a subfield F₀ if there exists an F₀-algebra A₀ such that A = A₀ ⊗F ₀ F. The following are shown, (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 F₀ of transcendence degree ≤ ≤1/2(n - 1) (n - 2) over k. (iii) If A is a ℤ/m × ℤ/2-crossed product for some m ≥ 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 F₀ such that trdeg _k(F₀) ≤ 4. (iv) If A has degree 4 then the trace form of A can be defined over a field F₀ of transcendence degree ≤ 4. (In (i), (iii) and (iv) it is assumed that the center of A contains certain roots of unity.).