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  "body": "\n## Definition\n\nA *differential field* is a [[field|0QA8CHH8MG]] $K$ with a map $\\partial \\colon K \\to K$ such that:\n\n1. $\\partial(a + b) = \\partial(a) + \\partial(b)$ for all $a, b \\in K$.\n2. $\\partial(ab) = \\partial(a) b + a \\partial(b)$ for all $a, b \\in K$.\n\nSo $\\partial$ is a [[derivation|R1F0HCCE6F]] of $K$ over $\\Z$.\nThe set $C = \\set{a \\in K : \\partial(a) = 0}$ is the *field of constants* of $(K, \\partial)$.\n\n## Remarks\n\nThe 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.\nExample: $K = \\C(x)$ 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.\nIts field of constants is $\\C$.\nExample: every field with $\\partial = 0$ is a differential field, with $C = K$.\nNon-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.\nA 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.\n",
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