## Rational Point

A -rational point is a point on an Algebraic Curve, where and are in a Field .

The rational point may also be a Point at Infinity. For example, take the Elliptic Curve and homogenize it by introducing a third variable so that each term has degree 3 as follows: Now, find the points at infinity by setting , obtaining Solving gives , equal to any value, and (by definition) . Despite freedom in the choice of , there is only a single Point at Infinity because the two triples ( , , ), ( , , ) 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.

The rational points on Elliptic Curves over the Galois Field GF( ) are 5, 7, 9, 10, 13, 14, 16, ... (Sloane's A005523).

Sloane, N. J. A. Sequence A005523/M3757 in An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html and Sloane, N. J. A. and Plouffe, S. The Encyclopedia of Integer Sequences. San Diego: Academic Press, 1995.