# Answer to Question #12456 in Abstract Algebra for john.george.milnor

Question #12456

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.

defined as p+q is third point of intersection or tangent line.

Expert's answer

(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.

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.

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