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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?

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

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    $\begingroup$ Please strive not to add more dupe answers to dupes of FAQs. $\endgroup$ – Bill Dubuque Mar 6 at 10:04

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