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  "body": "\n## Summary\n\nThe function field $k(X)$ of an algebraic variety $X$ over $k$ is the field of rational functions on $X$.\nA rational function is a function regular on some nonempty open subset.\nTwo of them are identified when they agree on a smaller open set.\nFor an affine variety it is the field of fractions of the coordinate ring.\nIt is a finitely generated field extension of $k$.\nIt is a birational invariant: two varieties have isomorphic function fields over $k$ exactly when they are birational.\n\n## Explanations\n\n### L1 human\n\nA rational function is a fraction of polynomials, like $x/y$, which makes sense except where the denominator vanishes.\nOn a variety, such a fraction is defined on a large open piece, and we do not care which piece.\nThe function field collects all of these fractions, and it is a field because every nonzero fraction can be turned upside down.\nThe function field of the affine line is $k(t)$, the field of rational functions in one variable.\n",
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