Differential field

Remarks

The field of constants is a subfield 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.