In number theory, a rational point is a point in space each of whose coordinates are rational; that is, the coordinates of the point are elements of the field of rational numbers.
For example, (3,−67/4) is a rational point in 2 dimensional space, since 3 and −67/4 are rational numbers. More generally, a K-rational point is a point on an algebraic variety where each coordinate of the point belongs to the field K. This means that, if the variety is given by a set of equations fj(x1, ..., xn) = 0, j=1, ..., m then the K-rational points are solutions (x1, ..., xn) ∈ Kn of the equations.
For example, the point P = (√2,3) is a point on the algebraic variety 3x2 − 2y = 0. Although P is not a rational point, since the coordinate √2 is not rational, P is an F-rational point, if F is chosen to be the field of numbers of the form a + b√2, where a and b are arbitrary rational numbers. This is because the coordinate √2 = 0 + 1√2, and the coordinate 3 = 3 + 0√2. In the parlance of morphisms of schemes, a K-rational point of a scheme X is just a morphism Spec K → X. The set of K-rational points is usually denoted X(K).
If a scheme or variety X is defined over a field k, a point x ∈ X is also called rational point if its residue field k(x) is isomorphic to k.
Rational points of varieties constitute a major area of current research.
For an abelian variety A, the K-rational points form a group. The Mordell-Weil theorem states that the group of rational points of an abelian variety over K is finitely generated if K is a number field.
The Weil conjectures concern the distribution of rational points on varieties over finite fields.