In Hartshorne exercise 2.5.14, he defines the homogeneous coordinate ring for projective schemes over an affine scheme. Then he specializes to the case where $X$ is a connected normal closed subscheme of $\mathbb P _k ^r$ where $k$ is an algebraically closed field. In the first part of this exercise, we have to show that $S$, the homogeneous coordinate ring of $X$, is a domain. I did this by showing that $X$ is integral and using the usual argument, namely Nullstellensatz, to deduce that $S$ is a domain. However it seemed to me that assuming $k$ to be algebraically closed is maybe unnecessarily strong, since I did not use this assumption afterwards (I solved this exercise).
I tried to come up with a counterexample where $S$ is not a domain (when $k$ is not algebraically closed), but it seems that I am not used to dealing with these stuff.
Is there a counterexample?