Algebraic closure in pseudofinite fields

Algebraic closure in pseudofinite fields

Thu, 14/06/2012
17:00 		Özlem Beyarslan (Bogazici) Logic Seminar Add to calendar L3
Algebraic closure in pseudofinite fields
A pseudofinite field is a perfect pseudo-algebraically closed (PAC) field which has $ \hat{\mathbb{Z}} $ as absolute Galois group. Pseudofinite fields exists and they can be realised as ultraproducts of finite fields. A group $ G $ is geometrically represented in a theory $ T $ if there are modles $ M_0\prec M $ of $ T $, substructures $ A,B $ of $ M $, $ B\subset acl(A) $, such that $ M_0\le A\le B\le M $ and $ Aut(B/A) $ is isomorphic to $ G $. Let $ T $ be a complete theory of pseudofinite fields. We show that, geometric representation of a group whose order is divisibly by $ p $ in $ T $ heavily depends on the presence of $ p^n $'th roots of unity in models of $ T $. As a consequence of this, we show that, for almost all completions of the theory of pseudofinite fields, over a substructure $ A $, algebraic closure agrees with definable closure, if $ A $ contains the relative algebraic closure of the prime field. This is joint work with Ehud Hrushovski.