The Differential Closure of a Differential Field

Good afternoon ladies and gentlemen. The subject of mathematical logic splits fourfold into: recursive functions, the heart of the subject; proof theory which includes the best theorem in the subject; sets and classes, whose romantic appeal far outweigh their mathematical substance; and model theory, whose value is its applicability to, and roots in, algebra. This afternoon I hope to sketch some theorems about differential fields first derived by model theoretic methods. In particular, I will indicate why every differential field si of characteristic 0 has a unique prime differentially closed extension called the differential closure of si. Model theory has proved useful in the study of differential fields because the notion of differential closure is surprisingly more complex than the analogous notions of algebraic closure, real closure, or Henselization. The virtue of model theory is its ability to organize succinctly the sort of tiresome algebraic details associated with elimination theory. The first concepts of model theory are structure and theory. Typic structures are groups, rings and fields. A theory is a set of sentences. A sentence is about the elements of some structure. The language of fields includes plus ( + ), times (•), equals (=) and variables that stand for elements of fields. A typic sentence in the language of fields says: every polynomial of degree 7 has a root. A typic theory is the theory of algebraically closed fields of characteristic 0 (ACF0). A structure si is said to be a model of a theory T if every sentence of T is true in si. Thus the models of ACF0 are what men call algebraically closed fields of characteristic 0. Pure model theory, at first thought, appears to be too general to have any mathematical substance. But that hasty thought is given the lie by several theorems, one of which is due to Vaught [1]: Let T be any coun-