---
tag: KPCN3KHG50
kind: concept
name: Differentially closed field
msc: [12H05, 03C60]
specializes: [GA5PTQGC6G]
sources:
  - url: https://en.wikipedia.org/wiki/Differentially_closed_field
same_as:
  wikidata: Q5275375
  wikipedia: Differentially closed field
---

## Summary

A differentially closed field is a [[differential field|GA5PTQGC6G]] in which every differential equation that can have a solution in some extension already has one.
It is the analogue, for differential equations, of an [[algebraically closed field|0PF6TM8TWT]] for polynomial equations.
In characteristic $0$, the differentially closed fields form an elementary class whose theory, called $\mathrm{DCF}_0$, is a central example in [[model theory|HFTBG5PMXC]].

## Explanations

### L1 human

In an algebraically closed field, every nonconstant polynomial has a root.
A differentially closed field asks the same of equations that involve an unknown $y$ and its derivatives $y', y'', \dots$.
Such fields are huge and abstract: no field of familiar functions, such as $\C(x)$, is differentially closed.
For instance, $\C(x)$ contains no solution of $y' = 1/x$, since $\log x$ is not a rational function.
Every differential field of characteristic $0$ sits inside a differentially closed one, much as every field sits inside an algebraically closed one.