The Group Law

Consider a cubic curve of the form $y^2=x^3+ax+b$ and assume that $x^3+ax+b$ has distinct roots. Then the set

$$E({\mathbb{R}}) = \{(x,y)\in {\mathbb{R}}\times {\mathbb{R}}: y^2 = x^3 + ax + b\}$$

is the graph of the real points on an elliptic curve. Given two solutions $(x_1,y_1)$ and $(x_2, y_2)$, there is a formula for a third solution $(x_3,y_3)$. It has the marvelous properties that

1. If $x_1,y_1,x_2,y_2\in{\mathbb{Q}}$ then $x_3,y_3\in{\mathbb{Q}}$.
2. The composition law turns the set $E({\mathbb{R}})$ into a GROUP.

The composition law is described in the text both algebraically and geometrically, but a complete proof that it has property 2 above is not given. I'm not sure what we'll do about this. My advice is that you would be best served to just believe this on faith at this point. When you learn ``algebraic geometry'' later in your career, you'll learn a beautiful and conceptually satisfying definition of the group law.