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Concept
Rational point
Summary
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 has no other rational points 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 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 k^n, that is, a sequence of n elements of k such that 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
are the pairs of rational numbers
where (a, b, c) is a Pythagorean triple.
The concept also makes sense in more general settings. A projective variety X in projective space \mathbb P^n over a field k can be defined by a collection of homogeneous polynomial equations in variables A k-point of \mathbb P^n, written is given by a sequence of n + 1 elements of k, not all zero, with the understanding that multiplying all of by the same nonzero element of k gives the same point in projective space. Then a k-point of X means a k-point of \mathbb P^n at which the given polynomials vanish.
More generally, let X be a scheme over a field k. This means that a morphism of schemes f: X → Spec(k) is given. Then a k-point of X means a of this morphism, that is, a morphism a: Spec(k) → X such that the composition fa is the identity on Spec(k).
Official source
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