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  "body": "\n## Summary\n\nThe perfect closure of a field $K$ of characteristic $p \u003e 0$ is the smallest [[perfect field|VWBVKT794W]] that contains $K$ inside an algebraic closure.\nIt is obtained by adjoining all $p^n$-th roots of all elements of $K$, so it is a [[purely inseparable extension|966GNVGB8A]] of $K$.\nIt is the purely inseparable counterpart of the [[separable closure|JJ91A1V8NQ]].\nFor a perfect field, and for any field of characteristic $0$, the perfect closure is the field itself.\n\n## Explanations\n\n### L1 human\n\nA field of characteristic $p$ is perfect when every element has a $p$-th root.\nIf some roots are missing, add them, then add their $p$-th roots, and so on forever.\nThe union of all these steps is the perfect closure.\nFor $\\F_p(t)$ one adds $t^{1/p}$, then $t^{1/p^2}$, and so on.\nThe result is the union of the fields $\\F_p(t^{1/p^n})$ for $n \\ge 0$.\n",
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