15.2 THE ELLIPTIC GROUP OVER THE REALS
The curve y2 = x3 − x = x(x − 1)(x + 1) enjoys a property characteristic of all elliptic curves – the chord-tangent group law discovered by Carl Gustav Jacob Jacobi (1804–1851) in the nineteenth century.
Proposition 15.1 (Bezout's Theorem): If P = (x1, y1) and Q = (x2, y2) are two points on the elliptic curve y2 = x3 + ax + b with 4a3 + 27b2 ≠ 0 and if the line PQ joining these points is not vertical, then the line PQ will intersect the curve in a third place ϕ(P, Q) = R = (x3, − y3) whose coordinates are given by
and
Proof: Suppose the equation of the line PQ is
There are two cases to consider; if x1 ≠ x2, then
Square y and substitute into the equation y2 = x3 + ax + b to obtain
so that
which gives
The case x1 = x2 is the limiting case ...
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