An AEC framework for fields with commuting automorphisms

Abstract In this paper, we introduce an AEC framework for studying fields with commuting automorphisms. Fields automorphisms are closely related to difference fields. Some authors define a ring (or field) as together several endomorphisms, while others only study one endomorphism. Z. Chatzidakis and E. Hrushovski have studied in depth the model theory of ACFA, companion automorphism. Our generalize setting. We they required commute. has proved that case two or more automorphisms, existentially closed models do not necessarily form first order class. present FCA-classes, prove FCA-class AP JEP thus monster model, Galois types coincide existential models, class is homogeneous, there version type amalgamation theorem allows combine three under certain conditions. Finally, use these results show our simple homogeneous structure sense S. Buechler O. Lessman (this non-elementary analogue classification theoretic notion theory).