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  "body": "\n## Definition\n\nLet $L/K$ be a [[field extension|0WQ1A8SHTZ]].\nThe extension $L/K$ is *purely transcendental* if $L = K(S)$ for a subset $S \\subseteq L$ that is [[algebraically independent|KV4PT6KW10]] over $K$.\n\n## Remarks\n\nThen $L$ is isomorphic over $K$ to the [[field of rational functions|4V81611P9N]] $K(x_s : s \\in S)$, by sending $x_s$ to $s$.\nExample: $\\Q(\\pi)/\\Q$ is purely transcendental, with $S = \\set{\\pi}$.\nNon-example: $\\Q(\\sqrt 2, t)/\\Q$ with $t$ transcendental is not purely transcendental.\nIndeed, $K$ is algebraically closed in $K(S)$ for independent $S$, but $\\sqrt 2$ is algebraic over $\\Q$ and not in $\\Q$.\nBy [[Lüroth's theorem|AVCZ6HVBA7]], every field strictly between $K$ and $K(t)$ is again purely transcendental over $K$.\n",
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