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  "body": "\n## Definition\n\nLet $L/K$ be an [[algebraic extension|09EYMKMSB9]].\nIf $K$ has [[characteristic|0P7CH1CND5]] $0$, the extension $L/K$ is *purely inseparable* if $L = K$.\nIf $K$ has characteristic $p \u003e 0$, the extension $L/K$ is *purely inseparable* if for every $\\alpha \\in L$ there is an integer $n \\ge 0$ with $\\alpha^{p^n} \\in K$.\n\n## Remarks\n\nIn characteristic $p \u003e 0$, the following are equivalent for an algebraic extension $L/K$:\n\n1. $L/K$ is purely inseparable;\n2. the [[minimal polynomial|0GK1WB9R98]] over $K$ of each $\\alpha \\in L$ has the form $x^{p^n} - a$ with $a \\in K$ and $n \\ge 0$;\n3. every element of $L$ that is separable over $K$ lies in $K$.\n\nExample: let $K = \\F_p(t)$ and $L = K(s)$ with $s^p = t$.\nCheck: every element of $L$ is $\\sum_{i=0}^{p-1} c_i s^i$ with $c_i \\in K$, and its $p$-th power is $\\sum c_i^p t^i \\in K$.\n\nNon-example: $\\F_{p^2}/\\F_p$ is not purely inseparable.\nFor $\\alpha \\in \\F_{p^2} \\setminus \\F_p$, the powers $\\alpha^{p^n}$ are $\\alpha$ or its conjugate $\\alpha^p$, and neither lies in $\\F_p$.\n",
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