---
tag: GA5PTQGC6G
kind: concept
name: Differential field
msc: [12H05]
specializes: [04C5ZBG1WT]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_algebra
---

## Summary

A differential field is a [[field|04C5ZBG1WT]] with a [[derivation|3J89REV9WQ]], an additive map that obeys the product rule, see the [[definition|T43TQ817XC]].
It is the algebraic setting for studying differential equations, where solutions are elements of larger differential fields.
The field $\C(x)$ of rational functions with $d/dx$ is the basic example.
[[Differential Galois theory|EEZS4H8QPH]] and the [[Liouville theorem|8VRTPECBPZ]] on elementary integrals are built on differential fields.

## Explanations

### L1 human

Take all fractions of polynomials in $x$, such as $(x^2 + 1)/(x - 3)$, together with the usual rule of differentiation.
This is a field where one can add, multiply, divide and also differentiate, and the derivative of every element is again in the field.
A differential field is any field with such a differentiation rule.
One can then ask whether an equation such as $y' = y$ has a nonzero solution in the field; here it does not, since the nonzero solutions $c e^x$ are not rational functions.