I've found in many textbooks that if $S$ is a smooth rational surface over an algebraically closed field, then $\chi(S,\mathcal{O}_S)=1$ (more precisely, $h^0=1$ and $h^1=h^2=0$).
I'm trying to find out if this is also true over arbitrary fields ($\Bbb{Q}$, for example), but I had difficulty to finding answers online.
If this is true, is there a standard reference for this kind of question?
If this is false in general, what's the reason behind it?
Thank you!