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  "body": "\n## Definition\n\nLet $(K, \\partial)$ be a [[differential field|T43TQ817XC]].\nLet $(L, \\partial)$ be a differential field that is a [[field extension|0WQ1A8SHTZ]] of $K$, with derivation equal to $\\partial$ on $K$.\nThe *differential Galois group* $\\Gal_\\partial(L/K)$ is the group, under composition, of the field automorphisms $\\sigma \\colon L \\to L$ such that:\n\n1. $\\sigma(a) = a$ for all $a \\in K$.\n2. $\\sigma(\\partial b) = \\partial(\\sigma b)$ for all $b \\in L$.\n\nDifferential Galois theory is the study of these groups.\nThe main case is a [[Picard–Vessiot extension|TB53Y5NMJD]] $L$ of $K$ for a linear differential equation.\nThere the field of constants of $K$ is assumed algebraically closed of characteristic $0$.\n\n## Remarks\n\nExample: let $K = \\C(x)$ with $\\partial = d/dx$ and $L = K(e^x)$, the Picard–Vessiot extension of $y' = y$.\nAn element $\\sigma$ of $\\Gal_\\partial(L/K)$ satisfies $\\partial(\\sigma e^x) = \\sigma(e^x)$, so $\\sigma(e^x)/e^x$ has derivative $0$ and is a nonzero constant $c \\in \\C$.\nEach $c \\in \\C^\\times$ gives such a $\\sigma$, because $e^x$ is transcendental over $K$, so $\\Gal_\\partial(L/K) \\cong \\C^\\times$.\nNon-example: let $K = \\C$ with $\\partial = 0$ and $L = \\C(x)$ with $\\partial = d/dx$; the field automorphism $\\sigma$ of $L$ over $\\C$ with $\\sigma(x) = 2x$ satisfies clause 1 but fails clause 2, since $\\partial(\\sigma x) = 2$ while $\\sigma(\\partial x) = 1$.\nThe elements of $\\Gal_\\partial(\\C(x)/\\C)$ are the maps $x \\mapsto x + c$ with $c \\in \\C$, so this group is isomorphic to $(\\C, +)$.\nFor a Picard–Vessiot extension for an equation of order $n$, the group acts on the $n$-dimensional space of solutions over the constants $C$ and is a Zariski-closed subgroup of $\\mathrm{GL}_n(C)$.\n",
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