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  "body": "\n## Definition\n\nLet $K$ be a [[field|0QA8CHH8MG]].\nThe field $K$ is *separably closed* if every [[separable polynomial|03J8BS7FMV]] $f \\in K[x]$ of degree at least $1$ has a root in $K$.\n\n## Remarks\n\nThe following are equivalent for a field $K$:\n\n1. $K$ is separably closed;\n2. every separable polynomial in $K[x]$ splits into linear factors over $K$;\n3. $K$ has no [[separable extension|0N019F6RE9]] of degree greater than $1$.\n\nEvery [[algebraically closed field|02SJG1HC32]] is separably closed.\nA field is algebraically closed if and only if it is separably closed and perfect.\n\nExample: the separable closure $K^{\\mathrm{sep}}$ of $K = \\F_p(t)$ is separably closed but not algebraically closed.\nCheck: the roots of a separable polynomial over $K^{\\mathrm{sep}}$ are separable over $K^{\\mathrm{sep}}$, hence over $K$, so they lie in $K^{\\mathrm{sep}}$.\nBut the root $t^{1/p}$ of $x^p - t$ is not separable over $K$.\n\nNon-example: a finite field $\\F_q$ is not separably closed.\nThe polynomial $x^q - x - 1$ is separable, since its derivative is $-1$, and it has no root in $\\F_q$, since $a^q - a - 1 = -1$ for every $a \\in \\F_q$.\n",
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