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  "body": "\n## Summary\n\nA purely transcendental extension of a field $K$ is an extension generated by [[algebraically independent|KV4PT6KW10]] elements.\nIt is isomorphic to a [[field of rational functions|4V81611P9N]] over $K$ in some set of variables.\nEvery field extension is an algebraic extension of a purely transcendental one, so these extensions are the building blocks of [[transcendental extensions|12JQCGTQ8W]].\n\n## Explanations\n\n### L1 human\n\nA purely transcendental extension adds free variables and nothing else.\nThe field $\\Q(x, y)$ of fractions of polynomials in $x$ and $y$ is the model example.\nAdding $\\sqrt 2$ as well gives an extension that is no longer purely transcendental, since $\\sqrt 2$ is tied to $\\Q$ by the equation $t^2 = 2$.\n",
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