With respect to the penultimate comment, does anyone have a reference for the fact that any (irreducible, which implies geometrically irreducible?) complex variety (finitely many polynomial equations in $n$ many variables, statement trivial when $n=1$) defined over the field of algebraic numbers (so coefficients can be assumed to be algebraic numbers) has at least one (or more?) point with values in the field of algebraic numbers (that is there's at least one solution with each coordinate an algebraic number)?