Differentially Closed Field - The Theory of Differentially Closed Fields

The Theory of Differentially Closed Fields

• p is 0 or a prime number, and is the characteristic of a field.
• A differential polynomial in x is a polynomial in x, ∂x, ∂2x, ...
• The order of a non-zero differential polynomial in x is the largest n such that ∂nx occurs in it, or −1 if the differential polynomial is a constant.
• The separant S_f of a differential polynomial of order n≥0 is the derivative of f with respect to ∂nx.
• The field of constants of a differential field is the subfield of elements a with ∂a=0.
• A field with derivation is called differentially perfect if the field of constants is perfect.
• A differentially closed field is a differentially perfect field K such that if f and g are differential polynomials such that S_f≠ 0 and g≠0 and f has order greater than that of g, then there is some x in the field with f(x)=0 and g(x)≠0. (Some authors add the condition that K has characteristic 0, in which case S_f is automatically non-zero, and K is automatically perfect.)
• DCF_p is the theory of differentially closed fields of characteristic p (0 or a prime).

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 DCF_p is complete and model complete (for p=0 this was shown by Robinson, and for p>0 by Wood (1973)). The theory DCF_p 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 Shelah (1973), and for κ countable by Hrushovski and Sokolovic.