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  "body": "\n## Summary\n\nA function field over a field $k$ is a finitely generated field extension of $k$.\nOne views it as the field of rational functions on some algebraic variety.\nIts [[transcendence degree|D7TR0YWR9H]] over $k$ is the number of variables, which is the dimension of the variety.\nFunction fields in one variable correspond to algebraic curves, and over finite fields they behave much like number fields.\n\n## Explanations\n\n### L1 human\n\nOn the plane, the rational functions are fractions of polynomials in $x$ and $y$.\nOn a curve such as $y^2 = x^3 + 1$, one uses the same fractions, but $y^2$ and $x^3 + 1$ count as equal.\nThe field of all such functions is the function field of the curve.\nMany questions about a curve, such as whether two curves are birational, become questions about its function field.\n",
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