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  "body": "\n## Summary\n\nA separably generated extension is a field extension with a [[transcendence basis|E45PA267S4]] over which the whole extension is [[separable|0WCX7NB4N7]] algebraic.\nSuch a basis is a separating transcendence basis.\nIt extends the notion of a separable algebraic extension to transcendental extensions.\nEvery extension in characteristic $0$, and every finitely generated extension of a [[perfect field|VWBVKT794W]], is separably generated.\n\n## Explanations\n\n### L1 human\n\nIn characteristic $p$, adding a $p$-th root can give an inseparable extension, where an irreducible polynomial has a repeated root.\nFor example, over $\\F_p(t)$ the polynomial $x^p - t$ is irreducible, but it equals $(x - t^{1/p})^p$ once $t^{1/p}$ is added.\nA separably generated extension is one where one can choose free variables so that the rest is reached by separable steps only.\nThe function field $\\F_p(x, y)$ with $y^2 = x^3 + 1$ and $p$ odd is separably generated over $\\F_p$, with separating basis $\\set{x}$.\n",
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