---
tag: T43TQ817XC
kind: definition
name: Differential field
msc: [12H05]
defines: GA5PTQGC6G
uses: [0QA8CHH8MG, R1F0HCCE6F, 07QY5RPFBR]
sources:
  - url: https://en.wikipedia.org/wiki/Differential_algebra
---

## Definition

A *differential field* is a [[field|0QA8CHH8MG]] $K$ with a map $\partial \colon K \to K$ such that:

1. $\partial(a + b) = \partial(a) + \partial(b)$ for all $a, b \in K$.
2. $\partial(ab) = \partial(a) b + a \partial(b)$ for all $a, b \in K$.

So $\partial$ is a [[derivation|R1F0HCCE6F]] of $K$ over $\Z$.
The set $C = \set{a \in K : \partial(a) = 0}$ is the *field of constants* of $(K, \partial)$.

## Remarks

The field of constants is a [[subfield|07QY5RPFBR]] of $K$: clauses 1 and 2 give $\partial(1) = 0$, and for $a \ne 0$ the rule $\partial(a^{-1}) = -\partial(a)/a^2$ holds.
Example: $K = \C(x)$ with $\partial = d/dx$ is a differential field; clauses 1 and 2 are the sum and product rules, and the derivative of a rational function is a rational function.
Its field of constants is $\C$.
Example: every field with $\partial = 0$ is a differential field, with $C = K$.
Non-example: any field with $\partial = \id$ fails clause 2, since $\partial(1 \cdot 1) = 1$ while $\partial(1) \cdot 1 + 1 \cdot \partial(1) = 2 \cdot 1$, and $1 \ne 2 \cdot 1$ in every field.
A field with finitely many commuting derivations is a partial differential field; a differential field in the sense above is also called an ordinary differential field.