Rational point
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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 larger fields that contain the rational numbers, such as the real numbers and the complex numbers.
For example, (3, −67/4) is a rational point in 2 dimensional space, since 3 and −67/4 are rational numbers. A special case of rational points are integer points, that is, when all of the coordinates are integers, e.g., (1, −5, 0) is an integral point in 3-dimensional space. On the other hand, 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 elements of larger fields containing the field K. This is analogous to rational points, which, as stated above, are contained in fields larger than the rationals. A corresponding special case of K-rational points are those that belong to a ring of algebraic integers existing inside the field K.
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[edit] Rational or K-rational points on algebraic varieties
If a K-rational point P exists on (that is, lies on) an algebraic variety, and the variety is given by a set of equations fj(x1, ..., xn)=0, j=1, ..., m, then P is an ordered n-tuple (x1, ..., xn) of numbers from the field K that is a solution of all of the equations simultaneously. 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.
[edit] 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.
It should be pointed out that there are also algebraic curves which contain just finitely many rational points (e.g. a line y=a x contains only one rational point (0, 0) for any irrational number a) or no rational points at all (e.g. a line y=x+b for any irrational number b).
[edit] Example 2
The point P=(√2, 3) is a point on the algebraic variety (in this case a parabola) given by the equation 3x2−2y=0. Although P is not a rational point, since the coordinate √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√2, where a and b are arbitrary rational numbers. This is because the coordinate √2=0+1√2, and the coordinate 3=3+0√2, and the numbers 0, 1, and 3 are rational.
[edit] Rational points of schemes
In the parlance of morphisms of schemes, a K-rational point of a scheme X is just a morphism Spec K→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∈X is also called a rational point if its residue field k(x) is isomorphic to k.
[edit] See also
- Algebraic curve
- Arithmetic dynamics
- Birational transformation
- Group of rational points on the unit circle
[edit] References
Rational points on Elliptic Curves, by Joseph H. Silverman and John Tate. Springer, 2010.
