Separable closure
= Separable closure
{title2=$K^{\mathrm{sep}}$}
A separable closure $K^{\mathrm{sep}}$ of a <field> $K$ is an algebraic separable extension that has no proper algebraic separable extension. Inside a chosen <algebraic closure> $\overline K$, it consists of the elements separable over $K$. Any two choices are isomorphic over $K$, although the isomorphism is not canonical.
In characteristic zero, every algebraic extension is separable, so a separable closure is an algebraic closure. For a finite field $\mathbb F_q$, it is the union of its finite extensions $\mathbb F_{q^n}$ inside an algebraic closure. Over an imperfect field, a separable closure need not be algebraically closed: purely inseparable algebraic elements may still be missing.
The automorphism group $\operatorname{Gal}(K^{\mathrm{sep}}/K)$ is the <absolute Galois group>. Its finite quotients organize finite Galois extensions of $K$. This is why separable closure, rather than an arbitrary large algebraic field, is the natural ambient field in <Galois theory>.