# Differentially closed field

In mathematics, a differential field *K* is **differentially closed** if every finite system of differential equations with a solution in some differential field extending *K* already has a solution in *K*. This concept was introduced by Robinson (1959). Differentially closed fields are the analogues
for differential equations of algebraically closed fields for polynomial equations.

## The theory of differentially closed fields

We recall that a differential field is a field equipped with a derivation operator. Let *K* be a differential field with derivation operator ∂.

- A
**differential polynomial**in*x*is a polynomial in the formal expressions*x*, ∂*x*, ∂^{2}*x*, ... with coefficients in*K*. - The
**order**of a non-zero differential polynomial in*x*is the largest*n*such that ∂^{n}*x*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 ∂^{n}*x*. - The
**field of constants**of*K*is the subfield of elements*a*with ∂*a*=0. - In a differential field
*K*of nonzero characteristic*p*, all*p*th powers are constants. It follows that neither*K*nor its field of constants is perfect, unless ∂ is trivial. A field*K*with derivation ∂ is called**differentially perfect**if it is either of characteristic 0, or of characteristic*p*and every constant is a*p*th power of an element of*K*. - A
**differentially closed field**is a differentially perfect differential 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*K*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**is the theory of differentially closed fields of characteristic_{p}*p*(where*p*is 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.

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 *p*th 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 DCF_{0} as its model completion.

The number of differentially closed fields of some infinite cardinality κ is 2^{κ}; for κ uncountable this was proved by Shelah (1973), and for κ countable by Hrushovski and Sokolovic.

## The Kolchin topology

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*.

## Quantifier elimination

Like the theory of algebraically closed fields, the theory DCF_{0} 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 DCF_{p} eliminates quantifiers in the language of differential fields with a unary function *r* added that is the *p*th root of all constants, and is 0 on elements that are not constant.

## Differential Nullstellensatz

The differential Nullstellensatz is the analogue in differential algebra of Hilbert's nullstellensatz.

- A
**differential ideal**or ∂-ideal is an ideal closed under ∂. - An ideal is called
**radical**if it contains all roots of its elements.

Suppose that *K* is a differentially closed field of characteristic 0. . Then Seidenberg's **differential nullstellensatz** states there is a bijection between

- Radical differential ideals in the ring of differential polynomials in
*n*variables, and - ∂-closed subsets of
*K*^{n}.

This correspondence maps a ∂-closed subset to the ideal of elements vanishing on it, and maps an ideal to its set of zeros.

## Omega stability

In characteristic 0 Blum () showed that the theory of differentially closed fields is ω-stable and has Morley rank ω. 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.

## See also

## References

- Marker, David (2000), "Model theory of differential fields",
*Model theory, algebra, and geometry*(PDF), Math. Sci. Res. Inst. Publ.,**39**, Cambridge: Cambridge Univ. Press, pp. 53–63, MR 1773702 - Robinson, Abraham (1959), "On the concept of a differentially closed field.",
*Bull. Res. Council Israel Sect. F*,**8F**: 113–128, MR 0125016 - Sacks, Gerald E. (1972), "The differential closure of a differential field",
*Bull. Amer. Math. Soc.*,**78**(5): 629–634, doi:10.1090/S0002-9904-1972-12969-0, MR 0299466 - Shelah, Saharon (1973), "Differentially closed fields",
*Israel J. Math.*,**16**(3): 314–328, doi:10.1007/BF02756711, MR 0344116 - Wood, Carol (1973), "The Model Theory of Differential Fields of Characteristic p ≠ 0",
*Proceedings of the American Mathematical Society*,**40**(2): 577–584, doi:10.2307/2039417, JSTOR 2039417 - Wood, Carol (1976), "The model theory of differential fields revisited",
*Israel Journal of Mathematics*,**25**(3–4): 331, doi:10.1007/BF02757008 - Wood, Carol (1998), "Differentially closed fields",
*Model theory and algebraic geometry*, Lecture Notes in Math.,**1696**, Berlin: Springer, pp. 129–141, doi:10.1007/BFb0094671, ISBN 978-3-540-64863-5, MR 1678539