Rational Point

A $K$-rational point is a point $(X,Y)$ on an Algebraic Curve, where $X$ and $Y$ are in a Field $K$.

The rational point may also be a Point at Infinity. For example, take the Elliptic Curve

$$Y^2 = X^3 + X + 42$$

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

$$ZY^2 = X^3 + XZ^2 + 42Z^3.$$

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

$$0 = X^3.$$

Solving gives $X=0$, $Y$ equal to any value, and (by definition) $Z=0$. Despite freedom in the choice of $Y$, 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 ($a$, $b$, 1) correspond to the ordinary points ($a$, $b$), and the triples ($a$, $b$, 0) correspond to the Points at Infinity, usually called the Line at Infinity.

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

See also Elliptic Curve, Line at Infinity, Point at Infinity

References

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.