Yet again, we see that the interplay between algebra and geometry leads to fascinating mathematics. In this case, we use geometry to construct the abelian group operation on the points of an elliptic curve. Then we derive algebraic formulas for this operation.
How to turn a nonsingular cubic curve into an abelian group is not that hard to describe geometrically. The hard part, which we will not perform fully, is verifying that all of the axioms described in chapter 7 are satisfied.
By now, we have arranged our definitions so that any line intersects the curve defined by a cubic ...