Rational points

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, as well as being elements of a larger field that contains the rational numbers, such as the real numbers or the complex numbers.

For example, $(3, -67/4)$ is a rational point in 2-dimensional space, because $3$ and $-67/4$ are rational numbers. A special case of a rational point is an integer point, that is, a point all of whose coordinates are integers. For example, $(1, -5, 0)$ is an integer point in 3-dimensional space. These are also called integral points.

More generally, a $K$ -rational point is a point in a space where each coordinate of the point belongs to the field $K$ , as well as being an elements of a larger field containing the field $K$ . A special case of $K$ -rational points are those that belong to a ring of algebraic integers existing inside the field $K$ .

Let $V$ be an algebraic variety over a field $K$ . $V$ is affine if it is given by a set of equations $f j (x 1, ..., x n) = 0$ , $j = 1, ..., m$ , with coefficients in $K$ . In this case, a $K$ -rational point $P$ of $V$ is an ordered n-tuple $(x 1, ..., x n)$ of elements of the field $K$ that is a solution of all of the equations simultaneously. In the general case, a $K$ -rational point of $V$ is a $K$ -rational point of some affine open subset of $V$ .

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