2 fixed a tag
1

# G-torsor whose ring of regular functions is a field.

I already asked this question on stackexchange but didn't get any answer. Maybe it is better suited for mathoverflow.

Let $G$ be an affine group scheme (not necessarily of finite type) over $\mathbb{Q}$. And let $P$ be a $G$-torsor (for the etale or fpqc topology) such that $H^0(P,\mathcal{O}_P)$ is a field.

Question: Is it possible for $H^0(P,\mathcal{O}_P)$ to be a field of transcendance degree $>0$ over $\mathbb{Q}$?

If $P$ is trivial torsor, then $H^0(P,\mathcal{O}_P) \simeq H^0(G,\mathcal{O}_G) \to \mathbb{Q}$ is an isomorphism.

If $P = Spec(K)$ is a finite Galois extension, then it is a torsor under $G = Spec(\mathbb{Q}^{Gal(K/\mathbb{Q})})$ i.e. $Gal(K/\mathbb{Q})$ seen as a $\mathbb{Q}$-group scheme.