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  "body": "\n## Definition\n\nLet $K$ be a field of [[characteristic|0P7CH1CND5]] $p \u003e 0$ and let $\\overline{K}$ be an [[algebraic closure|0MCXX29HWY]] of $K$.\nThe *perfect closure* of $K$ is the set\n$$K^{p^{-\\infty}} = \\set{\\alpha \\in \\overline{K} : \\alpha^{p^n} \\in K \\text{ for some integer } n \\ge 0}.$$\nFor a field $K$ of characteristic $0$, the perfect closure of $K$ is $K$.\n\n## Remarks\n\nThe set $K^{p^{-\\infty}}$ is a subfield of $\\overline{K}$, because $(\\alpha + \\beta)^{p^n} = \\alpha^{p^n} + \\beta^{p^n}$ in characteristic $p$.\nIt is a [[perfect field|MY8BG4EE0F]], it is [[purely inseparable|7FR2PZ141W]] over $K$, and it is contained in every perfect subfield of $\\overline{K}$ that contains $K$.\nIt is unique up to a unique isomorphism over $K$.\n\nExample: for $K = \\F_p(t)$, the perfect closure is $\\bigcup_{n \\ge 0} \\F_p(t^{1/p^n})$.\nCheck: each $\\F_p(t^{1/p^n})$ is purely inseparable over $K$, and the union is perfect because every element of $\\F_p(t^{1/p^n})$ has a $p$-th root in $\\F_p(t^{1/p^{n+1}})$.\n\nNon-example: for $p$ odd, the field $\\overline{K}$ is perfect and contains $K$, but it is not the perfect closure.\nIt contains $\\sqrt{t}$, and no power $\\sqrt{t}^{\\,p^n}$ lies in $K$, since $p^n$ is odd.\n",
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