Quote:
Originally Posted by Dr Sardonicus
Since A  B is an algebraic factor of A^n  B^n for every nonnegative integer n, we have that if f(x) is a polynomial in K[x], where K is a field, then
A  B is an algebraic factor of f(A)  f(B).
I imagine this has been known for centuries; I'm pretty sure Isaac Newton knew it, certainly for the cases where K is the rational or real numbers. Of course, the result continues to hold in cases where K is not a field, but I'm not sure offhand just how far you can push it. If K is a commutative ring (with 1) I don't see any reason it wouldn't work.
In particular, substituting x + k*f(x) for A and x for B, k*f(x) is an algebraic factor of f(x + k*f(x))  f(x).

Merely saying " I am pretty sure Isaac....." will not do; can you quote any paper or book where in either Newton, Euler or any mathematician has mentioned this result?