Rational point

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 $K$ .

Rational or K-rational points on algebraic varieties

Further information: Diophantine geometry

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 $V$ .

A variety $V$ is projective if it is defined in some projective space $P n$ by homogeneous polynomials $f 1, ..., f m$ (with coefficients in $K$ ). In this case, a $K$ -rational point of $V$ is a point $[x 0 : ... : x n]$ in the projective space, all of whose coordinates are in $K$ , which is a common solution of all the equations $f j = 0$ .

Sometimes, when no confusion is possible, or when $K$ is the field of the rational numbers, we say "rational points" instead of " $K$ -rational points".

Rational (as well as $K$ -rational) points that lie on an algebraic variety such as an elliptic curve constitute a major area of current research. For an abelian variety $A$ , the $K$ -rational points form a group. The Mordell–Weil theorem states that the group of rational points of an abelian variety over $K$ is finitely generated if $K$ is a number field.

The Weil conjectures concern the distribution of rational points on varieties over finite fields, where 'rational points' are taken to mean points from the smallest subfield of the finite field the variety has been defined over.

Examples

Example 1

The point $(3, - 67/4)$ is one of the infinite set of rational points on the straight line given by the equation $y + 67/4 = 2(x - 3)$ . This set of rational points forms a commutative group with group operation $(a, b) "+" (r, s) = (a + r, b + s + 91/4)$ , and group identity $(0, - 91/4)$ . It can be shown that there are no integral points on this particular line. This line is a simple type of an algebraic curve, which in turn is a type of algebraic variety. There are also algebraic curves that contain only finitely many rational points, or even no rational points at all. For example, the conic $x 2 + y 2 + 1 = 0$ has no rational points.

Example 2

The point $P = (\sqrt 2, 3)$ is a point on the parabola given by the equation $3 x 2 - 2 y = 0$ ; this is an example of an algebraic variety. Although $P$ is not a rational point, because the coordinate $\sqrt 2$ is not rational, $P$ is an $F$ -rational point, if $F$ is chosen to be the field of numbers of the form $a + b \sqrt2$ , where $a$ and $b$ are arbitrary rational numbers. This is because the coordinate $\sqrt 2 = 0 + 1 \sqrt 2$ , the coordinate $3 = 3 + 0\sqrt2$ , and the numbers $0$ , $1$ , and $3$ are rational.

Example 3

A point $(a, b, c)$ in the complex projective plane is $R$ -rational (or, as is common to say, is "real") if there exists a complex number $z$ such that $za$ , $zb$ and $zc$ are all real numbers. Otherwise, the point is a complex point. This description generalizes to complex projective spaces of higher dimension.

Rational points of schemes

In the language of morphisms of schemes, a $K$ -rational point of a scheme $X$ is just a morphism Spec $K \to X$ . The set of $K$ -rational points is usually denoted $X (K)$ .

If a scheme or variety $X$ is defined over a field $k$ , a point $x \in X$ is also called a rational point if its residue field $k (x)$ is isomorphic to $k$ .

References

Rational points on Elliptic Curves, by Joseph H. Silverman and John Tate. Springer, 2010.

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