Let $ G$ be a connected algebraic group over an algebraically closed field $ \overline{k}$ acting on an irreducible variety $ X$ . A geometric quotient is a morphism of varieties $ \pi: X \rightarrow X/\sim$ which on closed points (that is, as a morphism of classical varieties) satisfy the following:

(i): $ \pi$ is a surjective open map, and the fibres are exactly the $ G$ -orbits of $ X$ .

(ii): for any open set $ U$ of $ X/\sim$ , the ring homomorphism $ \pi^{\ast}: \overline{k}[U] \rightarrow \overline{k}[\pi^{-1}U]$ is an isomorphism onto the $ G$ -fixed points of codomain.

Rosenlicht’s theorem says that there exists a $ G$ -stable open subset of $ X$ for which the geometric quotient exists.

Is there any generalization of Rosenlicht’s theorem for when $ G$ and $ X$ are defined over an arbitrary field? The case I’m interested in is when $ G$ and $ X$ are geometrically connected subgroups of upper triangular unipotent matrices over a $ p$ -adic field $ F$ (so all orbits are closed), and $ X$ is normalized by $ G$ .