All monic polynomials of degree $d$ such that $f(x) | f(x^n) \forall n \in \mathbb{Z}^+$?

Question

The coefficients may be complex.

I was doing a problem for $d=4$ and am wondering if this can this problem be generalized for any $d$

Answer 1

$x$ root of $f(x)$ $\Rightarrow$ $x^n$ root of $f$ $\forall n$, but $f$ has only a finite number of roots $\Rightarrow$ $x=0$ or a $k-$th root of unit $\Rightarrow$ $f(x)$ is...

Answer 2

In fact one needs only the case $\,n=2\,$ to conclude that all zeros of $f$ are roots of unity. $ $ Generally, if $\,f(x)\in\Bbb C[x],\ f(0)\ne 0\,$ is such that $f(\alpha)=0\,\Rightarrow\, f(\alpha^2)=0$ or $f(-\alpha^2) = 0,\,$ then all zeros of $f$ are roots of unity. This leads to efficient algorithms to test if a polynomial is cyclotomic (i.e. divides $\,x^k-1\,$ for some $k),\,$ or a product of cyclotomic polynomials. Follow the above link for references and further discussion.