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  "body": "\n## Statement\n\nLet $(F, \\partial)$ be a [[differential field|GA5PTQGC6G]] of characteristic $0$, with [[derivation|3J89REV9WQ]] $\\partial$, and write $u' = \\partial u$.\nA differential [[field extension|0EQWVHF3W5]] $E = F(t_1, \\dots, t_m)$ of $F$ is *elementary* if, for each $i$, with $F_{i-1} = F(t_1, \\dots, t_{i-1})$, one of the following holds:\n\n1. $t_i$ is algebraic over $F_{i-1}$;\n2. $t_i' = u'/u$ for some nonzero $u \\in F_{i-1}$ (a logarithm);\n3. $t_i' = u' t_i$ for some $u \\in F_{i-1}$ (an exponential).\n\nLet $\\alpha \\in F$.\nSuppose that some elementary extension $E$ of $F$ with the same field of constants as $F$ contains an element $y$ with $y' = \\alpha$.\nThen there are constants $c_1, \\dots, c_n$ of $F$, nonzero elements $u_1, \\dots, u_n \\in F$, and an element $v \\in F$ such that\n$$\\alpha = v' + \\sum_{i=1}^{n} c_i \\frac{u_i'}{u_i}.$$\n\n## Remarks\n\nThe proof goes by induction on the length of the tower $F \\subseteq F_1 \\subseteq \\cdots \\subseteq E$.\nAt each step one writes the antiderivative in terms of the last generator $t_i$ and compares coefficients.\nThis works because derivatives of rational functions in a logarithm or an exponential have a controlled form.\nAn algebraic step is removed by taking traces, which is where the characteristic $0$ is used.\n\nExample: over $F = \\C(x)$ with $\\partial = d/dx$, the element $\\alpha = 1/x$ has the antiderivative $\\log x$, and indeed $\\alpha = c_1 u_1'/u_1$ with $c_1 = 1$, $u_1 = x$, $v = 0$.\nThe theorem is the tool that proves that $e^{-x^2}$, $e^x/x$ and $1/\\log x$ have no [[antiderivative|0BYWRWFT4F]] built from elementary functions.\n\nThe hypothesis on the constants cannot be dropped.\nOver $F = \\R(x)$, the element $\\alpha = 1/(1 + x^2)$ has the antiderivative $\\arctan x = \\frac{1}{2i} \\log \\frac{x - i}{x + i}$, which lies in an elementary extension that contains the new constant $i$.\nBut $\\alpha$ has no expression of the form above with real constants and $u_i, v \\in \\R(x)$.\n\n## Explanations\n\n### L1 human\n\nSome functions, such as $e^{-x^2}$, have no antiderivative that can be written with the usual functions: powers, roots, exponentials, logarithms and their combinations.\nLiouville's theorem explains why.\nIf an elementary antiderivative exists, it must have a very special shape: a function from the same field plus a sum of constant multiples of logarithms of such functions.\nOne can then check that no function of this shape works.\n\n### L2 human\n\nThe theorem turns the question \"is this integral elementary?\" into an algebraic question about the field $F$.\nFor $F = \\C(x, e^{g})$ with $g$ a nonconstant polynomial, it reduces to a first-order linear differential equation over $\\C(x)$.\nThis gives a classical criterion for rational $f$: the integral $\\int f e^{g} \\, dx$ is elementary exactly when $R' + g' R = f$ has a solution $R \\in \\C(x)$.\nThe [[Risch algorithm|VJ51B7J125]] makes this reduction systematic and decides elementary integrability in towers of logarithms, exponentials and algebraic functions.\nThe theorem is a basic result of differential algebra, alongside the theory of [[Picard–Vessiot extensions|TB53Y5NMJD]].\n",
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