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 , the Fermat curve of equation has no other rational points than , , and, if is even, and . Definition Given a field , and an algebraically closed extension of , an affine variety over is the set of common zeros in of a collection of polynomials with coefficients in : These common zeros are called the points of . A -rational point (or -point) of is a point of that belongs to , that is, a sequence of elements of such that for all . The set of -rational points of is often denoted . Sometimes, when the field is understood, or when is the field of rational numbers, one says "rational point" instead of "-rational point". For example, the rational points of the unit circle of equation are the pairs of rational numbers where is a Pythagorean triple. The concept also makes sense in more general settings. A projective variety in projective space over a field can be defined by a collection of homogeneous polynomial equations in variables A -point of written is given by a sequence of elements of , not all zero, with the understanding that multiplying all of by the same nonzero element of gives the same point in projective space. Then a -point of means a -point of at which the given polynomials vanish. More generally, let be a scheme over a field . This means that a morphism of schemes is given. Then a -point of means a section of this morphism, that is, a morphism such that the composition is the identity on . This agrees with the previous definitions when is an affine or projective variety (viewed as a scheme over ). When is a variety over an algebraically closed field , much of the structure of is determined by its set of -rational points. For a general field , however, gives only partial information about . In particular, for a