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

Logic