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  "body": "\n## Summary\n\nA purely inseparable extension is an [[algebraic extension|0PTBM7K0R2]] in which every element has some $p$-power, $p^n$, in the base field, where $p$ is the [[characteristic|0F6ES3BRYP]].\nIt is the opposite of a [[separable extension|0WCX7NB4N7]]: the only elements that are both separable and purely inseparable over $K$ are those of $K$.\nEvery algebraic extension splits into a separable part followed by a purely inseparable part, so the two kinds together describe all algebraic extensions.\nIn characteristic $0$ the only purely inseparable extension of $K$ is $K$ itself.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$, the map $x \\mapsto x^p$ respects sums and products.\nA purely inseparable extension adds elements that become old elements after you raise them to a $p$-th power a few times.\nFor example, over the field $\\F_p(t)$ of rational functions, you may add an element $s$ with $s^p = t$.\nThe polynomial $x^p - t$ then becomes $(x - s)^p$, so it has a single root repeated $p$ times.\nThis repeated root is what \"inseparable\" refers to.\n",
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