Taking g=1 and f any ordinary separable polynomial shows that any differentially closed field is separably closed. In characteristic 0 this implies that it is algebraically closed, but in characteristic p>0 differentially closed fields are never algebraically closed (or perfect), as the differential must vanish on any pth power. Unlike the complex numbers in the theory of algebraically closed fields, there is no natural example of a differentially closed field. Any differentially perfect field K has a differential closure, a prime model extension, which is differentially closed. Shelah showed that the differential closure is unique up to isomorphism over K. Shelah also showed that the prime differentially closed field of characteristic 0 (the differential closure of the rationals) is not minimal; this was a rather surprising result, as it is not what one would expect by analogy with algebraically closed fields.
The theory of DCFp is complete and model complete (for p=0 this was shown by Robinson, and for p>0 by ). The theory DCFp is the model companion of the theory of differential fields of characteristic p. It is the model completion of the theory of differentially perfect fields of characteristic p if one adds to the language a symbol giving the pth root of constants when p>0. The theory of differential fields of characteristic p>0 does not have a model completion, and in characteristic p=0 is the same as the theory of differentially perfect fields so has DCF0 as its model completion.
The number of differentially closed fields of some infinite cardinality κ is 2κ; for κ uncountable this was proved by , and for κ countable by Hrushovski and Sokolovic.
The Kolchin topology on K m is defined by taking sets of solutions of systems of differential equations over K in m variables as basic closed sets. Like the Zariski topology, the Kolchin topology is Noetherian.
A d-constructible set is a finite union of closed and open sets in the Kolchin topology. Equivalently, a d-constructible set is the set of solutions to a quantifier-free, or atomic, formula with parameters in K.
Like the theory of algebraically closed fields, the theory DCF0 of differentially closed fields of characteristic 0 eliminates quantifiers. The geometric content of this statement is that the projection of a d-constructible set is d-constructible. It also eliminates imaginaries, is complete, and model complete.
In characteristic p>0, the theory DCFp eliminates quantifiers in the language of differential fields with a unary function r added that is the pth root of all constants, and is 0 on elements that are not constant.
Suppose that K is a differentially closed field of characteristic 0. . Then Seidenberg's differential nullstellensatz states there is a bijection between
This correspondence maps a ∂-closed subset to the ideal of elements vanishing on it, and maps an ideal to its set of zeros.
|last=Sacks|first= Gerald E. |title=The differential closure of a differential field. |journal=Bull. Amer. Math. Soc.|volume= 78 |year=1972|pages= 629-634 |url=http://www.ams.org/bull/1972-78-05/S0002-9904-1972-12969-0/}} |last=Shelah|first= Saharon |title=Differentially closed fields. |journal=Israel J. Math.|volume= 16 |year=1973|pages= 314-328 |doi=10.1007/BF02756711}} |last=Wood|first= Carol |chapter=Differentially closed fields|title= Model theory and algebraic geometry|pages=129-141 |series=Lecture Notes in Math.|volume= 1696|publisher= Springer|place= Berlin|year= 1998 |doi=10.1007/BFb0094671}}