Rational Point (original) (raw)

Algebra Applied Mathematics Calculus and Analysis Discrete Mathematics Foundations of Mathematics Geometry History and Terminology Number Theory Probability and Statistics Recreational Mathematics Topology

Alphabetical Index New in MathWorld

A -rational point is a point (X,Y) on an algebraic curve f(X,Y)=0 , where and are in a field . For example, rational point in the field of ordinary rational numbers is a point (X,Y) satisfying the given equation such that both and are rational numbers.

The rational point may also be a point at infinity. For example, take the elliptic curve

	(1)

and homogenize it by introducing a third variable so that each term has degree 3 as follows:

	(2)

Now, find the points at infinity by setting Z=0 , obtaining

	(3)

Solving gives X=0 , equal to any value, and (by definition) Z=0 . Despite freedom in the choice of, there is only a single point at infinity because the two triples ( X_1 , Y_1 , Z_1 ), ( X_2 , Y_2 , Z_2 ) are considered to be equivalent (or identified) only if one is a scalar multiple of the other. Here, (0, 0, 0) is not considered to be a valid point. The triples (, , 1) correspond to the ordinary points (, ), and the triples (,, 0) correspond to the points at infinity, usually called the line at infinity.