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  "body": "\n## Summary\n\nA differential field is a [[field|04C5ZBG1WT]] with a [[derivation|3J89REV9WQ]], an additive map that obeys the product rule, see the [[definition|T43TQ817XC]].\nIt is the algebraic setting for studying differential equations, where solutions are elements of larger differential fields.\nThe field $\\C(x)$ of rational functions with $d/dx$ is the basic example.\n[[Differential Galois theory|EEZS4H8QPH]] and the [[Liouville theorem|8VRTPECBPZ]] on elementary integrals are built on differential fields.\n\n## Explanations\n\n### L1 human\n\nTake all fractions of polynomials in $x$, such as $(x^2 + 1)/(x - 3)$, together with the usual rule of differentiation.\nThis is a field where one can add, multiply, divide and also differentiate, and the derivative of every element is again in the field.\nA differential field is any field with such a differentiation rule.\nOne can then ask whether an equation such as $y' = y$ has a nonzero solution in the field; here it does not, since the nonzero solutions $c e^x$ are not rational functions.\n",
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