A groupoid on a field is just a groupoid scheme (U, R, s, t, c) where U = Spec(k) with k a field.
Groupoids on fields are very similar to groupschemes. Many results on group schemes have analogues for groupoids on fields. I recently added a bunch of new results on these types of objects to the stacks project. The goal was to see to whether I could now prove: If s, t are finite type, does there exist a finite index subfield k_0 of k such that s, t agree as morphisms to Spec(k_0).
Now, it turns out that this is a somewhat tricky thing to prove from the material we have in the stacks project so far. But looking into the future this is really quite straightforward. Let me explain.
[Hypothetical Argument] Suppose more generally that (U, R, s, t, c) is a groupoid scheme with s, t flat, of finite presentation whose stabilizer group scheme G is flat and locally of finite presentation over U. Then G acts freely on R and the quotient R’ = R/G is an algebraic space (future result). Then R’ is an equivalence relation on U with s’, t’ : R’ —> U flat and locally of finite presentation. So U/R’ is an algebraic space (future result). In fact, U/R’ is the coarse moduli space of [U/R] and actually [U/R] —> U/R’ is a gerbe (banded by G somehow).
Going back to U = Spec(k), s, t finite type, the spectrum of the field k_0 is going to be the coarse moduli space in the previous paragraph. The index of k_0 in k will be finite as a gerbe acquires a point over a finite extension (+ small argument).
The hypothetical argument above (hypothetical in that I haven’t written out all the details, and some hypotheses might need to be added) can also be used as the basis for decomposing any algebraic stack with suitable finiteness assumptions into gerbes over algebraic spaces. In the language of groupoid schemes: given (U, R, s, t, c) and say everybody finite type over a Noetherian base find a stratification U = \bigcup U_i such that the restriction of the stabilizer group scheme G to U_i is flat.