Some model theory of separably closed fields

The model theory of separably closed fields was first investigated by Ersov. Among other things he proved that the first-order theory of separably closed fields of a fixed characteristic p / 0 and of fixed degree of imperfection e £ ω U {00} is complete, see [6]. In 1979 C. Wood (see [24]) showed that these theories are stable, but not superstable, yielding the only examples of stable, non-superstable fields. Further model theoretic properties of these fields, like quantifier elimination, equationality, the independence relation, DOP, etc. were analysed. In 1988 F. Delon (see [5]) published a comprehensive article in which she investigated types in terms of their associated ideals in an appropriate polynomial ring, in particular proving elimination of imaginaries and giving a detailed analysis of different notions of rank. In 1992, E. Hrushovski gave a model theoretic proof of the Mordell-Lang conjecture for function fields. In the case of characteristic p φ 0 he used some of the model theoretic tools for separably closed fields, in particular an analysis of minimal types and the author's results on definability in separably closed fields. A separably closed field can be equipped with a differential structure. Accounts of this line of work can be found in [21,22,23,10]. The purpose of these notes is to give an overview of the known results in the model theory of separably closed fields with special emphasis on the case of finite degree of imperfection. When discussing elimination of imaginaries, we give a general outline of how this property can be proved in all known examples of stable fields with additional structure