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  "body": "\n## Definition\n\nLet $k$ be a [[field|0QA8CHH8MG]] and let $n \\ge 0$ be an integer.\nA *function field in $n$ variables over $k$* is a [[field extension|0WQ1A8SHTZ]] $K/k$ with two properties.\nFirst, $K = k(a_1, \\dots, a_m)$ for finitely many $a_1, \\dots, a_m \\in K$.\nSecond, the [[transcendence degree|D7TR0YWR9H]] of $K/k$ is $n$.\n\n## Remarks\n\nMany authors also ask that $k$ be algebraically closed in $K$, and some ask $n \\ge 1$.\nExample: the [[field of rational functions|4V81611P9N]] $k(x_1, \\dots, x_n)$ is a function field in $n$ variables.\nExample: if $k$ has characteristic not $2$, the field $k(x)[y]/(y^2 - x^3 - 1)$ is a function field in one variable, with [[transcendence basis|E45PA267S4]] $\\set{x}$; it is the function field of the curve $y^2 = x^3 + 1$.\nNon-example: the field $k(x_1, x_2, x_3, \\dots)$ in infinitely many variables is not finitely generated over $k$.\nFor an irreducible variety $V$ over $k$, the field $k(V)$ of rational functions on $V$ is a function field in $\\dim V$ variables over $k$.\n",
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