We introduce a new device in the study of abstract elementary classes (AECs): Galois Morleyization, which consists in expanding the models of the class with a relation for every Galois (orbital) type of length less than a fixed cardinal κ. We show: Theorem 0.1 (The semantic-syntactic correspondence). An AEC K is fully (< κ)-tame and type short if and only if Galois types are syntactic in the Ga...

This paper introduces a natural extension of Kolchin’s differential Galois theory to positive characteristic iterative differential fields, generalizing to the non-linear case the iterative Picard-Vessiot theory recently developed by Matzat and van der Put. We use the methods and framework provided by the model theory of iterative differential fields. We offer a definition of strongly normal ex...