I have to admit I’ve been using somewhat unconventional definitions. The usual definition of an affine variety is as an **irreducible** Zariski-closed subset of , affine -space over an algebraically closed field . A generic Zariski-closed subset is usually referred to instead as an algebraic set (although some authors also call these varieties), and the terminology does not apply to non-algebraically closed fields. The additional difficulty that arises in the non-algebraically-closed case is that it’s harder to think about points. For example, has two types of points corresponding to the two types of irreducible polynomials: the usual points on the real line and additional points . These points can be thought of as orbits of the action of on , hence can be thought of as the quotient of by this group action. This picture generalizes.

Anyway, for convenience let’s stick to . In this case, and more generally in the algebraically closed case, there is a reasonably simple description of what the category of affine varieties looks like, but first we have to describe what the morphisms look like and then we have to take the strong Nullstellensatz on faith, since we haven’t proven it yet.