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  "body": "\n## Definition\n\nLet $L/K$ be an [[algebraic extension|09EYMKMSB9]].\nAn element $\\alpha \\in L$ is separable over $K$ if its [[minimal polynomial|0GK1WB9R98]] over $K$ has no repeated root in an algebraic closure.\nThe *separable closure of $K$ in $L$* is the set $L_s$ of elements of $L$ that are separable over $K$.\nLet $\\overline{K}$ be an [[algebraic closure|0MCXX29HWY]] of $K$.\nThe *separable closure* of $K$ is the separable closure $K^{\\mathrm{sep}}$ of $K$ in $\\overline{K}$.\n\n## Remarks\n\nThe set $L_s$ is a subfield of $L$ containing $K$, the extension $L_s/K$ is [[separable|0N019F6RE9]], and every separable subextension of $L/K$ lies in $L_s$.\nThe extension $L/L_s$ is purely inseparable.\nThe field $K^{\\mathrm{sep}}$ is separably closed, and it is unique up to an isomorphism over $K$.\n\nExample: let $K = \\Q$ and $L = \\overline{\\Q}$.\nCheck: in characteristic $0$ every irreducible polynomial has distinct roots, so every element of $\\overline{\\Q}$ is separable and $\\Q^{\\mathrm{sep}} = \\overline{\\Q}$.\n\nNon-example: let $K = \\F_p(t)$ and $L = K(s)$ with $s^p = t$.\nThe element $s$ is not in the separable closure of $K$ in $L$, since its minimal polynomial $x^p - t = (x - s)^p$ has a repeated root; in fact $L_s = K$.\n",
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