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 $G$.



Rosenlicht's theorem and rationality questions
Jim Humphreys was right: No generalization is necessary. In Theorem 2 of


[Rosenlicht, Maxwell: Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956) 401--443](https://www.jstor.org/stable/2372523?seq=1#page_scan_tab_contents),


the result is already stated over arbitrary base fields.


