Differentially closed field

Definition

Let $(K, \partial)$ be a differential field of characteristic $0$ with one derivation $\partial$ . Let $K\{y\}$ be the ring of differential polynomials over $K$ . It is the polynomial ring $K[y, y', y'', \dots]$ in the variables $y^{(k)}$ , $k \ge 0$ , with $\partial$ extended by $\partial(y^{(k)}) = y^{(k+1)}$ . The order of $f \in K\{y\}$ is the largest $k$ such that $y^{(k)}$ occurs in $f$ ; a nonzero element of $K$ has order $-1$ . For $a \in K$ , $f(a)$ is the element obtained by replacing each $y^{(k)}$ by $\partial^k a$ . Let $f, g \in K\{y\}$ be nonzero, with the order of $g$ less than the order of $f$ . The field $K$ is differentially closed if, for all such $f$ and $g$ , there is $a \in K$ with $f(a) = 0$ and $g(a) \ne 0$ .

Remarks

Taking $f$ of order $0$ , that is $f \in K[y]$ nonconstant, and $g = 1$ shows that a differentially closed field is algebraically closed. Its field of constants $\set{c \in K : \partial c = 0}$ is also algebraically closed. Every differential field of characteristic $0$ is contained in a differentially closed field, and it has a differential closure, which is unique up to isomorphism over it. The theory $\mathrm{DCF}_0$ of differentially closed fields of characteristic $0$ is complete, has elimination of quantifiers and is $\omega$ -stable.

Example: by the existence result above, the differential closure of $(\Q, 0)$ is a differentially closed field. Differentially closed fields are known through such existence results rather than by explicit construction.

Non-example: $\C(x)$ with $\partial = d/dx$ is not differentially closed. Take $f = x y' - 1$ , of order $1$ , and $g = 1$ , of order $-1$ . A solution would be $a \in \C(x)$ with $a' = 1/x$ , and no rational function has this derivative.

Non-example: a field with the zero derivation is not differentially closed, since $f = y' - 1$ has no zero in it.

Tag	`B2YPDYMR7E`
URI	`m2://B2YPDYMR7E`
Kind	definition
Writing	W0 stub
MSC	12H05, 03C60
Defines	Differentially closed field
Uses	Differential field, Derivation, Algebraically closed field
Changed	2 November 2026

Differentially closed field

Definition

Remarks

Reading path

Further reading

What this page needs for W1

What links here