separable closure

Inside an algebraic closure of a field, some elements are 'well-behaved' (separable) and some, in positive characteristic, are not. The separable closure collects exactly the well-behaved ones: it is the largest piece of the algebraic closure built entirely from separable elements. Over a perfect field it coincides with the full algebraic closure, but over imperfect fields it is a strictly smaller subfield, and it is the natural arena for Galois theory.

Precisely, fix an algebraic closure K-bar of K. The separable closure K_sep is the set of all elements of K-bar that are separable over K, i.e. whose minimal polynomial has distinct roots. This is in fact a field, it is the maximal separable subextension of K-bar over K, and the extension K_sep over K is separable while K-bar over K_sep is purely inseparable.

The separable closure carries the absolute Galois group of K: the automorphism group Gal(K_sep / K) is a profinite group whose study is central to number theory and Galois cohomology. Over characteristic 0 or finite fields the separable closure equals the algebraic closure, so the distinction is invisible there; it only matters over imperfect fields like F_p(t), where K-bar properly contains K_sep.