Coincidently, I thought about this a few weeks ago (without any conclusion). I think that I can prove that a "weakly locally separated" algebraic space *X* with an étale cover Spec(*k*)->*X* is of the form Spec(*k'*) if *X* lives over a field *k*_0 such that *k*/*k*_0 is algebraic. If *X* is not locally separated, this condition does not always hold (take **A**^1/**Z** where **Z** acts by translation and restrict this action to the generic point).

Let *K* be the algebraic closure of *k*. Let *R*=Spec(*K*) x_ *X* Spec(*K*). By assumption

j : *R* -> Spec(*K*) x_{*k*_0} Spec(*K*)

is an immersion and it is enough to show that this is a closed immersion (since fpqc morphisms descend closed immersions).

We can replace *k*_0 with its perfect closure. This follows from the observation that *R* is reduced.

Now, the right-hand scheme is a group scheme over Spec(*K*). Indeed, it is the fundamental group scheme \pi_0(*k*_0). It is totally disconnected and all its residue fields are *K* and the group of *K*-points is the pro-finite group Gal(*K*/*k*_0).

*R* is also totally disconnected and all its points have residue field *K*. The map

j(*K*) : *R*(*K*) -> Gal(*K*/*k*_0)

is injective and locally closed. Since *R*(*K*) => *K*(*K*) is an equivalence relation, it follows that j(*K*) identifies *R*(*K*) with a subgroup of Gal(*K*/*k*_0).

**Lemma**: A locally closed subgroup *H* of a topological group *G* is closed.

pf: The closure of *H* is a subgroup so we can assume that *H* is open. It is then easily seen that the complement of *H* is open.

Thus, *R*(*K*) is a *closed* subgroup of Gal(*K*/*k*_0). In particular, *j* is a closed immersion.

**Remark**: If *K*/*k*_0 is not algebraic, then (if *K* is algebraically closed) we still have a group structure on the *K*-points of the fiber product of *K* over *k*_0. *R*(*K*) will be a closed subgroup of this group but it is not clear whether this implies that *j* is closed.