Monic polynomials with simple roots problem

[annarodriguez]

How can we find all the monic polynomials P(X) ∈ C [X], with simple roots such that P(Xⁿ) = ±P(X) P(ζX) P (ζ²X) …P (ζ^n-1X), where ζ is a primitive n-th root of unity?

 

[annarodriguez]

How can we find all the monic polynomials P(X) ∈ C [X], with simple roots such that P(Xⁿ) = ±P(X) P(ζX) P (ζ²X) …P (ζ^n-1X), where ζ is a primitive n-th root of unity?

I was thinking that the roots of P(X) are included into the set of roots of the equation xⁿ=x, but I'm honestly stuck on how I could prove this...