variety over a field and any field extension of , also determines the set of -rational points of , meaning the set of solutions of the equations defining with values in . Example: Let be the conic curve in the affine plane over the real numbers Then the set of real points is empty, because the square of any real number is nonnegative. On the other hand, in the terminology of algebraic geometry, the algebraic variety over is not empty, because the set of complex points is not empty. More generally, for a scheme over a commutative ring and any commutative -algebra , the set of -points of means the set of morphisms over . The scheme is determined up to isomorphism by the functor ; this is the philosophy of identifying a scheme with its functor of points. Another formulation is that the scheme over determines a scheme over by base change, and the -points of (over ) can be identified with the -points of (over ). The theory of Diophantine equations traditionally meant the study of integral points, meaning solutions of polynomial equations in the integers rather than the rationals For homogeneous polynomial equations such as the two problems are essentially equivalent, since every rational point can be scaled to become an integral point. Rational points on curves Much of number theory can be viewed as the study of rational points of algebraic varieties, a convenient setting being smooth projective varieties. For smooth projective curves, the behavior of rational points depends strongly on the genus of the curve. Genus 0 Every smooth projective curve of genus zero over a field is isomorphic to a conic (degree 2) curve in If has a -rational point, then it is isomorphic to over , and so its -rational points are completely understood. If is the field of rational numbers (or more generally a number field), there is an algorithm to determine whether a given conic has a rational point, based on the Hasse principle: a conic over has a rational point if and only if it has a point over all completions of that is, over and all p-adic fields Genus 1 It is harder to determine whether a curve of genus 1 has a rational point. The Hasse principle fails in this case: for example, by Ernst Selmer, the cubic curve in has a point over all completions of but