turkmath.org

This is joint work with Piotr Kowalski. A G-field is a field, together with an action of a group G by field automorphisms. Our purpose is to give an axiomatization of the theory of "generic", i.e. existentially closed G-fields. If such axiomatization for the class of existentially closed G-fields exists, we call the resulting theory G-TCF. If G is the trivial group then G-TCF is the theory of algebraically closed fields, ACF. If G is the group of integers, then G-TCF exists and its theory is very well studied ACFA, the theory of algebraically closed fields with a generic automorphism. It is also known that G-TCF exist if G is finite, and G is a finitely generated free group. A natural generalization of finite groups and free groups is the class of virtually free groups. Our main theorem says that, when G is a finitely generated virtually free group, then G-TCF exists. We also give field theoretic propertied of G-fields.