Contents

Abstract. 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 F0 if there exists an F0-algebra A0 such that A = A0⊗F0 F. 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 F0 of transcendence degree ≤ 12 (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, Mm(A) can be defined over a field F0 such that trdegk(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) we assume that the center of A contains certain roots of unity.