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.
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.
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.
In characteristic 0 Blum showed that the theory of differentially closed fields is ω-stable and has Morley rank ω.
[citation needed] In non-zero characteristic Wood (1973) showed that the theory of differentially closed fields is not ω-stable, and Shelah (1973) showed more precisely that it is stable but not superstable.