In Hartshorne chapter I.6 is discussed the construction of the abstract nonsingular curve as part of the proof for the well known correspondence between complete regular irreducible algrebaic curves and the function fields in one variable, in the case of algebraically closed ground field.
It seems that Milne (in his notes on Modular Forms, theorem 2.7, pag. 96) propose a same construction for the general case (i.e. non algebraically closed ground field), but I couldn't find any reference for the details of this proof in such case. Is there any?