Definition of Elliptic Curves

I think it is about time we defined what an elliptic curve is. It is a nonsingular cubic curve with a distinguished point, usually denoted by . We say that the elliptic curve is defined over a field if the curve is defined over and .

It turns out that for every field , the set can be endowed with a structure of an abelian group as follows.

Bezout’s Theorem

We are coming across this theorem so often, that I think it deserves a post of its own. We keep wishing that the number of intersection of two curves defined by equation of degrees and respectively, be but we constantly hit snags. In this post, we try to iron out things out all the way.