 Separable closure

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 $\mathrm{Gal}(K^{\mathrm{sep}}/K)$ 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.