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  "body": "\n## Summary\n\nThe field of rational functions $K(x)$ over a field $K$ consists of the quotients $p/q$ of polynomials with $q \\ne 0$.\nIt is the [[field of fractions|0F2PTGX1A2]] of the [[polynomial ring|0SDC934R52]] $K[x]$.\nIt is the simplest field that contains $K$ and an element $x$ that is not a root of any nonzero polynomial over $K$.\nIt is the basic example of a [[purely transcendental extension|G5PVJMM37Z]] and of a [[function field|XKEXQDRTGH]].\n\n## Explanations\n\n### L1 human\n\nPolynomials can be added, subtracted and multiplied, but not always divided.\nAllowing fractions of polynomials, as with fractions of integers, gives a field.\nFor example, $\\frac{1}{x}$ and $\\frac{x^2 + 1}{x - 1}$ lie in $\\Q(x)$.\nTwo fractions are equal when cross multiplication gives the same polynomial, as for $\\frac{x}{x^2} = \\frac{1}{x}$.\nThe [[definition|JX6CHG7HWV]] makes this precise.\n",
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