ABSTRACT
C l a im . For n sufficiently large, if do,. . . , an are independent realizations of tp(a/E), then E C aclCT(ao,. . . , an).
and by assumption
Corollary 4.22 then gives
§5. Study of the fixed field. A particularly important definable subset of a model K of ACFA is the fixed field, Fix(a) = {x e K \ o(x) = x}. It turns out that this subfield is responsible for much of the bad behavior of models of ACFA. We will show that Fix(cr) is a pseudofinite field, i.e., is elementarily equivalent to an ultraproduct of finite fields (or equivalently, is an infinite
model of the theory of finite fields). Pseudofinite fields were first studied by Ax, see [9]. Some very nice results on pseudofinite fields were also obtained in two papers by E. Hrushovski and A. Pillay ([11] and [12]).
