Let $G$ be an affine group scheme over $\mathbb{Q}$. Then it is easy to see that if the ring of regular functions $H^0(G,\mathcal{O}_G)$ is a field then $G$ is the trivial group.
Let $P$ be a $G$-torsor (for the etale or fpqc topology). Is it possible for $H^0(P,\mathcal{O}_P)$ to be a field of transcendance degree $>0$?