I am try to prove that $X(C) = \{\text{nilpotent elements in C}\}$ is not a scheme.
I have proved that it is local, so what makes it is not a scheme must be the open cover stuff. And I think it is covered by $X_n(C) = \{x|\text{$x^n=0$ in C}\}$. So I think it is covered by an infinite union of affine schemes. Therefore, I think the problem is that the cover is not open.
But even if I could prove this obvious cover is not open. It is still not sufficient to prove that $X$ has no open affine cover. And I have no idea for finding some contradiction assuming $X$ has an open affine cover.
So how to show that $X$ has no open affine cover? Thanks for any help.