Rational point

In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. In number theory and algebraic geometry, a rational point of an algebraic variety is a point whose coordinates belong to a given field. If the field is not mentioned, the field of rational numbers is generally understood. If the field is the field of real numbers, a rational point is more commonly called a real point. Understanding rational points is a central goal of number theory and Diophantine geometry. For example, Fermat's Last theorem may be restated as: for n > 2, the Fermat curve of equation x n + y n = 1 {displaystyle x^{n}+y^{n}=1} has no other rational point than (1, 0), (0, 1), and, if n is even, (–1, 0) and (0, –1). Given a field k, and an algebraically closed extension K of k, an affine variety X over k is the set of common zeros in K n {displaystyle K^{n}} of a collection of polynomials with coefficients in k: These common zeros are called the points of X. A k-rational point (or k-point) of X is a point of X that belongs to kn, that is, a sequence (a1,...,an) of n elements of k such that fj (a1,...,an) = 0 for all j. The set of k-rational points of X is often denoted X(k). Sometimes, when the field k is understood, or when k is the field Q of rational numbers, one says 'rational point' instead of 'k-rational point'. For example, the rational points of the unit circle of equation