K-rational point

In number theory and algebraic geometry, a rational point of an algebraic variety is a solution of a set of polynomial equations in a given field. If the field is not mentioned, the field of rational numbers may be understood. For example, (3/5, 4/5) is a rational point on the circle x² + y² = 1. Understanding rational points is a central goal of number theory and specifically of diophantine geometry.

An affine variety X in affine n-space Aⁿ over a field k is defined by a collection of polynomial equations with coefficients in k:

A k-rational point (or k-point) of affine space Aⁿ means an element of the product set kⁿ, that is, a sequence (a₁,...,a_n) of n elements of k. A k-rational point of X means a k-rational point of Aⁿ such that f_j(a₁,...,a_n) = 0 in k for all j. The set of k-rational points of X is called X(k).

The concept also makes sense in more general settings. A projective variety X in projective space Pⁿ over a field k can be defined by a collection of homogeneous polynomial equations in variables x₀,...x_n. A k-point of Pⁿ, written [a₀,...,a_n], is given by a sequence of n+1 elements of k, not all zero, with the understanding that multiplying all of a₀,...a_n 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 Pⁿ at which the given polynomials vanish.

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