Polynomials such that $P(X)\mid P(X^2)$

Question

I just found this problem in an old email but I have forgotten how to do it.

Find the number of monic irreducible polynomials $P \in \mathbb{Z}[X]$ such that $P(X) \mid P\left(X^2\right)$ and $\deg(P) = 144$.

Answer 1

If $r$ is a root of $P(X)$, it must also be a root of $P(X^2)$, which says $r^2$ is a root of $P(X)$. Thus squaring maps the set of roots of $P(X)$ into itself. This implies that all roots of $P(X)$ must be either $0$ or roots of unity. $P(X)$ is either $X$ or a cyclotomic polynomial. Moreover, you can show that only cyclotomic polynomials $\Phi_j(X)$ with odd $j$ work. Now, what odd $j$ have $\Phi_j(X)$ of degree $144$?