Answer to Question #12456 in Abstract Algebra for john.george.milnor
defined as p+q is third point of intersection or tangent line.
Every element has its inverse:
inverse to (x, x^3) is (-x, -x^3), as third point of intersection of line
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,
if equation of this line is y=kx+b, then
equation has 3 real roots.
By Vietes theorem: a+b+c=(coef.
So, abscice for sum of points is a+b=-c. This implies
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