| Résume | We take a look at structures that consist of a field together with finitely many distinguished field automorphisms required to commute. The theory of fields with one distinguished automorphism has a model companion known as ACFA, which Z. Chatzidakis and E. Hrushovski have studied in depth. However, Hrushovski has proved that if you look at fields with two or more commuting automorphisms, then the existentially closed models of the theory do not form a first order model class. This leads us to investigate them within a non-elementary framework. One way of doing non-elementary model theory is to move from elementary classes to the more general setting of abstract elementary classes (AECs). In the first order world, classes of structures are usually defined syntactically as model classes of a given first order theory. An AEC is defined more semantically, as a class of structures together with a binary relation that generalises the first-order elementary submodel relation. In this talk, we go through some basics of AECs and present an AEC framework for studying fields with commuting automorphisms. |
