Basic question: Global sections of structure sheaf is a field 
$X$ is projective and reduced over a field $k$ (not necessarily algebraically closed). Why is $H^0(X,\mathcal{O}_X)$ a field?

Are there any good lecture notes on this (valuative criteria, properness, projectiveness, completeness)? I really don't have time to go through EGA/SGA.

This is not true. The scheme $X=\mathrm{Spec}(k\times k)$ is projective and reduced over $k$ but the global sections of the structure sheaf do not form a field. You need to assume $X$ is connected too. In any case, by properness, $H^0(X,\mathscr{O}_X)$ is a finite-dimensional $k$-algebra (this is the hard part), hence a finite product of Artin local rings. Since $X$ is connected, $H^0(X,\mathscr{O}_X)$ is a connected ring, meaning that it must be a single Artin local ring. The unique maximal ideal of such a ring coincides with its nilradical, which is zero because $X$ is reduced, so $H^0(X,\mathscr{O}_X)$ must be a field. 
