Find all monic complex polynomials $P(x)$ such that $P(x)|P(x^2)$.
My progress so far is that I have find that for degree 1, $P(x)=x, x^2$ are the only ones.
For degree 2, they are $P(x)=x^2+x+1, x^2, x^2-1, x^2-x, x^2-2x+1$.
I also prove that these are only solutions for degree 1 and 2. However I do not see how this generalizes. Any help please?