Equational theories of fields

Abstract

A complete first-order theory is equational if every definable set is a Boolean combination of instances of equations, that is, of formulae such that the family of finite intersections of instances has the descending chain condition. Equationality is a strengthening of stability. We show the equationality of the theory of proper extensions of algebraically closed fields of some fixed characteristic and of the theory of separably closed fields of arbitrary imperfection degree. Srour showed that the theory of differentially closed fields in positive characteristic is equational. We give also a different proof of his result.