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  "body": "\n## Definition\n\nLet $(K, \\partial)$ be a [[differential field|GA5PTQGC6G]] of characteristic $0$ with one [[derivation|3J89REV9WQ]] $\\partial$.\nLet $K\\{y\\}$ be the ring of differential polynomials over $K$.\nIt is the polynomial ring $K[y, y', y'', \\dots]$ in the variables $y^{(k)}$, $k \\ge 0$, with $\\partial$ extended by $\\partial(y^{(k)}) = y^{(k+1)}$.\nThe *order* of $f \\in K\\{y\\}$ is the largest $k$ such that $y^{(k)}$ occurs in $f$; a nonzero element of $K$ has order $-1$.\nFor $a \\in K$, $f(a)$ is the element obtained by replacing each $y^{(k)}$ by $\\partial^k a$.\nLet $f, g \\in K\\{y\\}$ be nonzero, with the order of $g$ less than the order of $f$.\nThe field $K$ is *differentially closed* if, for all such $f$ and $g$, there is $a \\in K$ with $f(a) = 0$ and $g(a) \\ne 0$.\n\n## Remarks\n\nTaking $f$ of order $0$, that is $f \\in K[y]$ nonconstant, and $g = 1$ shows that a differentially closed field is [[algebraically closed|0PF6TM8TWT]].\nIts field of constants $\\set{c \\in K : \\partial c = 0}$ is also algebraically closed.\nEvery differential field of characteristic $0$ is contained in a differentially closed field, and it has a differential closure, which is unique up to isomorphism over it.\nThe theory $\\mathrm{DCF}_0$ of differentially closed fields of characteristic $0$ is complete, has elimination of quantifiers and is $\\omega$-stable.\n\nExample: by the existence result above, the differential closure of $(\\Q, 0)$ is a differentially closed field.\nDifferentially closed fields are known through such existence results rather than by explicit construction.\n\nNon-example: $\\C(x)$ with $\\partial = d/dx$ is not differentially closed.\nTake $f = x y' - 1$, of order $1$, and $g = 1$, of order $-1$.\nA solution would be $a \\in \\C(x)$ with $a' = 1/x$, and no rational function has this derivative.\n\nNon-example: a field with the zero derivation is not differentially closed, since $f = y' - 1$ has no zero in it.\n",
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