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  "body": "\n## Definition\n\nLet $L/K$ be an [[algebraic extension|09EYMKMSB9]].\nAn element $a \\in L$ is separable over $K$ if its [[minimal polynomial|0GK1WB9R98]] over $K$ is a [[separable polynomial|03J8BS7FMV]].\nThe extension $L/K$ is separable if every element of $L$ is separable over $K$.\n\n## Remarks\n\nAn irreducible polynomial over $K$ is separable if and only if its derivative is not the zero polynomial.\nSo every algebraic extension of a field of characteristic $0$ is separable.\nEvery algebraic extension of a finite field is separable.\nFor a prime $p$ and the field $K = \\mathbb{F}_p(t)$ of rational functions, the extension $K(t^{1/p})/K$ is not separable: the minimal polynomial $x^p - t$ equals $(x - t^{1/p})^p$.\n\n## Explanations\n\n### L1 human\n\nIn a separable extension, the roots of each minimal polynomial are all different.\nProblems arise only in characteristic $p$, where a derivative like that of $x^p - t$ can vanish.\n",
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