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  "body": "\n## Statement\n\nLet $K$ be a [[field|0QA8CHH8MG]] and let $K(t)$ be the [[field of rational functions|4V81611P9N]] in one variable $t$ over $K$.\nLet $E$ be a field with $K \\subsetneq E \\subseteq K(t)$.\nThen $E = K(u)$ for some [[rational function|0WCZH4AFMT]] $u \\in K(t) \\setminus K$.\n\n## Remarks\n\nPick $u \\in E \\setminus K$ and write $u = f(t)/g(t)$ with $f, g$ coprime polynomials.\nThen $t$ is a root of $f(x) - u\\, g(x)$ over $K(u)$, and one shows that $[K(t) : K(u)] = \\max(\\deg f, \\deg g)$.\nThe proof then takes the minimal polynomial of $t$ over $E$, chooses $u$ among its coefficients not in $K$, and compares degrees to get $E = K(u)$.\nExample: $E = K(t^2 + t^{-2})$ lies between $K$ and $K(t)$, and the theorem says it is already of the form $K(u)$; here $u = t^2 + t^{-2}$ works, and $[K(t) : E] = 4$.\nThe analogue in two variables concerns fields $E \\subseteq K(x, y)$ with $K(x, y)$ finite over $E$.\nIt holds over algebraically closed fields of characteristic $0$ by Castelnuovo's theorem, and it fails in general in three or more variables.\n\n## Explanations\n\n### L1 human\n\nEvery field between $K$ and the rational functions in one variable is again a field of rational functions in one variable.\nSo one cannot find an exotic field squeezed in there.\nGeometrically, a curve that is the image of the line under a rational map is itself rational.\n\n### L2 human\n\nThe theorem says that a subfield of a [[purely transcendental extension|G5PVJMM37Z]] of transcendence degree $1$ is purely transcendental.\nIn geometry, it gives that a unirational curve is rational.\nThe degree formula $[K(t) : K(u)] = \\max(\\deg f, \\deg g)$ also computes the degree of a rational map from the line to itself, which is the [[degree|0GS74FGFEK]] of the field extension $K(t)/K(u)$.\n",
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