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  "body": "\n## Definition\n\nLet $k$ be an algebraically closed field and let $X$ be an [[algebraic variety|HV6YBS0YDH]] over $k$.\nConsider pairs $(U, f)$ where $U \\subseteq X$ is a nonempty [[open subset|02BH3MCBQB]] and $f$ is a [[regular function|4M94FZH9Y8]] on $U$.\nTwo pairs $(U, f)$ and $(V, g)$ are equivalent if $f = g$ on $U \\cap V$.\nThe *function field* $k(X)$ of $X$ is the set of equivalence classes, called *rational functions* on $X$, with the operations\n$$[(U, f)] + [(V, g)] = [(U \\cap V, f + g)], \\qquad [(U, f)] \\cdot [(V, g)] = [(U \\cap V, fg)].$$\n\n## Remarks\n\nSince $X$ is irreducible, $U \\cap V$ is nonempty and dense, and two regular functions that agree on a nonempty open subset of $U \\cap V$ agree on all of $U \\cap V$.\nThis makes the relation transitive and the operations well defined.\n$k(X)$ is a field: if $f \\ne 0$ on $U$, then $1/f$ is regular on the nonempty open set $U \\setminus V(f)$.\n\nIf $X$ is an [[affine variety|JERVQT4C12]], then $k(X)$ is the [[field of fractions|0F2PTGX1A2]] of the [[coordinate ring|G9WB27VWP7]] $k[X]$.\nIf $U$ is a nonempty open subset of $X$, then $k(U) = k(X)$.\n$k(X)$ is a [[function field|XKEXQDRTGH]] over $k$ in the algebraic sense: a finitely generated field extension of $k$.\n\nExample: $k(\\mathbb{A}^n) = k(x_1, \\dots, x_n)$, the [[field of rational functions|4V81611P9N]], since $k[\\mathbb{A}^n]$ is the polynomial ring.\nExample: for the parabola $X = V(y - x^2)$, $k[X] \\cong k[x]$, so $k(X) \\cong k(x)$.\n\nNon-example: the construction needs irreducibility.\nIn $X = V(xy) \\subseteq k^2$, the open sets $\\set{x \\ne 0}$ and $\\set{y \\ne 0}$ of $X$ are nonempty and disjoint.\nSo the relation above is not transitive there, and the analogous ring built from $k[X] = k[x, y]/(xy)$, its total ring of fractions, is $k(x) \\times k(y)$, which is not a field.\n",
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