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M. Lorenz, Z. Reichstein, L. H. Rowen, D. J. Saltman *,
Fields of definition for division algebras*, Journal
of the London Mathematical Society, 68, Issue 03 (2003), 651-670.

**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.

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