I'm trying to solve exercise 7.5.11 from Lovett's "Abstract Algebra":
Suppose that $p\mid n$. Prove that $\Phi_{pn}(x)=\Phi_n(x^p)$.
Here, $\Phi_n(x)$ is the $n$th cyclotomic polynimial and I don't think $p$ is restricted to prime numbers. I fixed an integer $p\geq 1$ and tried to prove that $\Phi_{p^{k+1}}(x)=\Phi_{p^k}(x^p)$ for all integer $k\geq 1$. I tried using mathematical induction on $k$ but I couldn't prove that it holds when $k=1$.