is π1(Spec k), the étale fundamental group of a field k. Essentially by definition, the fundamental group of k can be shown to be isomorphic to the absolute Galois group Gal (ksep / k). More precisely, the choice of a geometric point of Spec (k) is equivalent to giving a separably closed extension field K, and the étale fundamental group with respect to that base point identifies with the Galois group Gal (K / k). This interpretation of the Galois group is known as Grothendieck's Galois theory. More generally, for any geometrically connected variety X over a field k (i.e., X is such that Xsep := X ×k ksep is connected) there is an exact sequence of profinite groups 1 → π1(Xsep, ) → π1(X, ) → Gal(ksep / k) → 1. Schemes over a field of characteristic zero For a scheme X that is of finite type over C, the complex numbers, there is a close relation between the étale fundamental group of X and the usual, topological, fundamental group of X(C), the complex analytic space attached to X. The algebraic fundamental group, as it is typically called in this case, is the profinite completion of π1(X). This is a consequence of the Riemann existence theorem, which says that all finite étale coverings of X(C) stem from ones of X. In particular, as the fundamental group of smooth curves over C (i.e., open Riemann surfaces) is well understood; this determines the algebraic fundamental group. More generally, the fundamental group of a proper scheme over any algebraically closed field of characteristic zero is known, because an extension of algebraically closed fields induces isomorphic fundamental groups. Schemes over a field of positive characteristic and the tame fundamental group For an algebraically closed field k of positive characteristic, the results are different, since Artin–Schreier coverings exist in this situation. For example, the fundamental group of the affine line is not topologically finitely generated. The tame fundamental group of some scheme U is a quotient of the usual fundamental group of U which takes into account only covers that are tamely ramified along D, where X is some compactification and D is the complement of U in X. For example, the tame fundamental group of the affine line is zero. Affine schemes over a field of characteristic p It turns out that every affine scheme is a -space, in the sense that the