Is there a relation between polynomials $f(a)$ with $f(b)$ and $f(ab)$?

Question

Let $f(x) = c_n x^n + \dots + c_1x + c_0$ be a polynomial of degree $n$ with integer coefficients.

Is there a relation between $f(a)$ with $f(b)$ and $f(ab)$ in general?

For example: $f(ab) = f(a)f(b) \big( \cfrac{1}{c_n + \dots + c_0} \big)$.

Answer 1

I'm not sure, but I just want to say that sometimes it's very difficult to work from some generalization backwards to identify some relationship, even though it might look cooler to say it the way you're saying it - ask yourself the first couple cases and check it out

• Case n = 0

$f(a) = c_{a0} \\ f(b) = c_{b0}\\ f(ab) = c_{ab0}$

• Case n = 1

$f(a) = c_{a1}a + c_{a0} \\ f(b) = c_{b1}b + c_{b0}\\ f(ab) = c_{ab1}ab + c_{ab0}$

etc

then you can start messing around with them - maybe look at some morphisms, throw them in some rings, I don't know - this math world is your oyster :)