Question

Show that set of points (x,x^3) where x is any real form abelian group under + operation
defined as p+q is third point of intersection or tangent line.

Answer 1

(0,0) point will be zero of this group.
Every element has its inverse:
inverse to (x, x^3) is (-x, -x^3), as third point of intersection of line
through them
is origin.
Commutativity is obvious, as for line it doesn't
matter what point to go through first, and what second.

If some line
intersects our curve in three points with first coordinate a,b,c respectively,
then
if equation of this line is y=kx+b, then
kx+b=x^3
and last
equation has 3 real roots.
So x^3-kx-b=0
By Vietes theorem: a+b+c=(coef.
near x^2)=0
So, abscice for sum of points is a+b=-c. This implies
associativity.