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  "body": "\n## Definition\n\nLet $R$ be a [[commutative ring|0GYGHKX834]] and let $f \\in R[x]$.\nThe polynomial $f$ is separable if there are $a, b \\in R[x]$ with $a f + b f' = 1$, where $f'$ is the [[formal derivative|0AD4VS02H3]] of $f$.\n\n## Remarks\n\nOver a [[field|0QA8CHH8MG]] $K$, $f$ is separable if and only if $f$ has no repeated root in an [[algebraic closure|0MCXX29HWY]] of $K$.\nThe polynomial $f = x^2 - 2$ is separable over $\\mathbb{Q}$, because $f' = 2x$ and $-\\tfrac{1}{2} f + \\tfrac{1}{4} x f' = 1$.\nOver $\\mathbb{F}_p(t)$, the irreducible polynomial $x^p - t$ is not separable, because its derivative is $0$.\nSome older texts call an irreducible polynomial separable when it has distinct roots, and extend this to other polynomials factor by factor.\n\n## Explanations\n\n### L1 human\n\nA separable polynomial has distinct roots, in a field that is large enough to contain all of them.\nOver the rational numbers, every irreducible polynomial is separable.\nInseparable irreducible polynomials only appear in prime characteristic.\n",
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