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Old 2017-06-01, 06:05   #6
devarajkandadai's Avatar
May 2004

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Originally Posted by Dr Sardonicus View Post
Since A - B is an algebraic factor of A^n - B^n for every non-negative 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?
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