A Constructive Definition of Points on an Algebraic Curve

Heuristically, an algebraic curve is the locus of Pointspoints in the xy-plane for which $$\chi (x,y) = 0$$ χ ( x , y ) = 0 for some irreducible polynomial $$\chi (x,y)$$ χ ( x , y ) with rational coefficients. However, from a geometric point of view, this conception of a “curve” is clearly unsatisfactory. First, x and y must both be allowed to have complex values in order for this locus to represent a plausible image of the corresponding “curve,” which means that the xy-plane is not a plane at all, but a 4-dimensional manifold, in which the curve is a 2-dimensional submanifold. Moreover, the inclusion of “points at infinity” is well known to simplify interpretations of many geometric properties of “curves,” and such points can be integrated into the notion of a locus only by further artificial constructions. In this book, the heuristic interpretation of curves as loci will remain in the background, and the theorems and proofs will deal instead with the field of rational functions on the curve, which is a strictly algebraic construction.