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  "body": "\n## Definition\n\nLet $(K, \\partial)$ be a [[differential field|GA5PTQGC6G]] of characteristic $0$, where $\\partial$ is a [[derivation|3J89REV9WQ]] of $K$.\nLet $C = \\set{c \\in K : \\partial c = 0}$ be its field of constants, and assume that $C$ is [[algebraically closed|0PF6TM8TWT]].\nLet\n$$L(y) = y^{(n)} + a_{n-1} y^{(n-1)} + \\cdots + a_1 y' + a_0 y$$\nwith $a_0, \\dots, a_{n-1} \\in K$, where $y^{(k)}$ denotes $\\partial^k y$.\nA *Picard–Vessiot extension* of $K$ for $L$ is a differential [[field extension|0EQWVHF3W5]] $E \\supseteq K$ such that:\n\n1. There are $y_1, \\dots, y_n \\in E$ with $L(y_i) = 0$ for each $i$ that are linearly independent over the constants of $E$.\n2. $E$ is generated as a field over $K$ by $y_1, \\dots, y_n$ and all their derivatives, that is, $E = K(y_i^{(k)} : 1 \\le i \\le n, \\ 0 \\le k \\le n - 1)$.\n3. The field of constants of $E$ is $C$.\n\n## Remarks\n\nUnder these hypotheses a Picard–Vessiot extension for $L$ exists, and it is unique up to an isomorphism of differential fields over $K$.\nThe group of automorphisms of $E$ over $K$ that commute with $\\partial$ is the differential Galois group of $L$; it is a linear algebraic group over $C$.\n\nExample: take $K = \\C(x)$ with $\\partial = d/dx$, so $C = \\C$, and $L(y) = y' - y$.\nThe field $E = \\C(x, e^x)$ is a Picard–Vessiot extension: $y_1 = e^x$ solves $L(y) = 0$, it generates $E$ over $K$, and the constants of $\\C(x, e^x)$ are only the complex numbers.\n\nNon-example: for the same $K$ and $L$, the field $E' = \\C(x, e^x, e^{x^2})$ contains the solution $e^x$ and has constant field $\\C$, but it fails clause 2.\nIt is strictly larger than the field generated by the solutions and their derivatives, because $e^{x^2}$ does not lie in $\\C(x, e^x)$.\n\nThe extension is an analogue of the [[splitting field|0KXQKX9P63]] of a polynomial, and it is the basic object of [[differential Galois theory|EEZS4H8QPH]].\n",
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