Publication status: appeared in J. London Math. Soc. 68 (2003), 651 - 670; arXiv: math.RA/0110198

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 of F if A can be obtaind by extension of scalars from some F₀-subalgebra A₀ 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 ³ 5, then A is defined over a field F₀ of transcendence degree at most (n-1)(n-2)/2 over k.
3. If A is a Z/m ×Z/2-crossed product for some m ³ 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) is at most 4.
4. If A has degree 4 then the trace form of A can be defined over a field F₀ of transcendence degree at most 4.

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