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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $K[x]$ be the [[polynomial ring|0Q9EZS90GY]] in one variable $x$ over $K$.\nThe field of rational functions $K(x)$ is the [[field of fractions|0VF11ZTTMD]] of $K[x]$.\nIts elements are the fractions $p/q$ with $p, q \\in K[x]$ and $q \\ne 0$, where $p/q = p'/q'$ if and only if $p q' = p' q$.\nMore generally, for variables $x_1, \\dots, x_n$, the field $K(x_1, \\dots, x_n)$ is the field of fractions of $K[x_1, \\dots, x_n]$.\n\n## Remarks\n\nExample: $\\frac{x^2 - 1}{x - 1} = \\frac{x + 1}{1}$ in $\\Q(x)$, since $(x^2 - 1) \\cdot 1 = (x + 1)(x - 1)$.\nNon-example: no element $p/q$ of $\\Q(x)$ satisfies $(p/q)^2 = x$, since $p^2 = x q^2$ would give $2 \\deg p = 1 + 2 \\deg q$, which is impossible.\nSo $K(x) \\supset K$ is a field extension in which $x$ is [[transcendental|0NQMZ89X4A]] over $K$.\nAn element of $K(x)$ is a formal fraction, not a function on $K$: over $\\F_2$, the fraction $\\frac{1}{x^2 + x}$ is a nonzero element, although $x^2 + x$ vanishes at every point of $\\F_2$.\nOver $\\R$ or $\\C$, the elements correspond to [[rational functions|0WCZH4AFMT]] in the sense of analysis.\n\n## Explanations\n\n### L1 human\n\nYou build $K(x)$ from polynomials in the same way as you build the rational numbers from the integers.\nEvery nonzero fraction of polynomials has an inverse, obtained by turning it upside down.\n",
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