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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 $>0$?

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    Not even a comment or an up vote? Is something wrong with the question?2012-08-12

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