I am trying to follow the proof in the book Abstract Algebra by Dummit and Foote (Theorem 41, pg. 554) that $\Phi_n$ is an irreducible monic polynomial in $\mathbb{Z}[x]$ of degree $\varphi(n)$.
What I understand is that if it is not irreducible, than we can factor $\Phi_{n}=fg$ where $f$ is the minimal polynomial of $\zeta_{n}$ (a primitive $n$-th root of unity).
I also agree that if $p$ is a prime s.t. $(p,n)=1$ then $f(\zeta_{n}^{p})=0$. The question is why if $(a,n)=1$ then $f(\zeta_{n}^{a})=0$ (I don't understand why this is true even for $a=p_{1}p_{2}$).
Can someone please explain why we can move from primes to their product?