Let be an elliptic curve defined over . We have seen that a line with rational slope passing through one rational point on need not intersect in rational points only. But if passes through two rational points on , then the third intersection point must be rational. This is because a cubic polynomial with two rational roots must have all its roots rational. One caveat is required, however: in order to be guaranteed to have three intersection points, one must interpret intersections in the sense of Bezout's Theorem; in other words, one really should work in the projective plane over the complex numbers, and count intersection points with multiplicities. It turns out that among all the points at infinity in the projective plane, only one is on the elliptic curve; i.e., the line at infinity intersects only in one point (with multiplicity 3, though!)
For an example, let us go back to the elliptic curve of the previous section with equation . Let us find the third intersection point of with the line through and . The equation of is , so the -coordinates of the points in are solutions to
Using this operation of taking two rational points and producing a third, we can develop a way to ``add'' two points. One says that the set of rational points on can be given the structure of an abelian group. This means that there is an operation that takes two rational points on and produces a new rational point on , such that the following axioms are satisfied:
The specific addition rule on the elliptic curve is characterized by the following rules:
Note: a line passes through if and only if it is vertical or is the ``line at infinity.''
As an example, let us compute , where and . We already found that the line intersects in the points , , and . Therefore, by Rule 2, . Thus . The vertical line intersects in the three points , and , so . Hence , so .
In general, the recipe for adding two points and on an elliptic curve is as follows: draw the line through and . (If , draw the line tangent to at , in order to get a line that intersects at with multiplicity at least 2.) Find a third point such that consists of , , and (listed with multiplicity if necessary). If , then equals ; if , equals the reflection of in the -axis.