Thread: A question of history View Single Post 2017-05-30, 13:16   #5
Dr Sardonicus

Feb 2017
Nowhere

2×5×7×73 Posts Quote:
 Originally Posted by devarajkandadai I may be wrong perhaps I am the first mathematician to discover the following property of polynomials: Let f(x) be a polynomial in x ( x belongs to Z, can be a Gaussian integer, or be a square matrix in which the elements are rational integers or Gaussian integers). Then f(x + k*f(x)) = = 0 mod(f(x)).
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).  