# Finding all monic complex polynomials $P(x)$ such that $P(x)|P(x^2)$ [duplicate]

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?

• Oh yes, thank you pointing that out. It is because any working polynomial can be scaled to monic. I has edited it in. – jxia1234 Mar 5 at 23:28

Hint: If $$P(c)=0$$ then $$0=P(c^{2})=P(c^{4})=P(c^{8})=...$$ and a polynomial can only have finite number of roots. Can you take over from here?