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  "body": "\n## Summary\n\nA Picard–Vessiot extension is the smallest [[differential field|GA5PTQGC6G]] extension that contains a full set of solutions of a linear differential equation.\nIt adds no new constants.\nIt plays the role for linear differential equations that the splitting field plays for polynomial equations.\nIts group of differential automorphisms is the differential Galois group of the equation, the main object of [[differential Galois theory|EEZS4H8QPH]].\n\n## Explanations\n\n### L1 human\n\nStart with a field of functions, such as the rational functions $\\C(x)$, and a [[linear differential equation|0QQRP17Z7S]] with coefficients in it.\nOften the solutions are not in the field, so one adjoins them together with all their derivatives.\nThe result is a bigger field in which the equation can be solved completely.\nFor $y' = y$ over $\\C(x)$, one adjoins $e^x$ and gets the field $\\C(x, e^x)$.\nThe symmetries of this bigger field that fix $\\C(x)$ and commute with differentiation tell how the solutions are related to each other.\n",
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