How to show that $\Phi_{p^n}(X)=\Phi_p(X^{p^{n-1}})$ with $p$ prime where $\Phi_n(X)$ is the cyclotomic polynomial define by $$\Phi_n(X)=\prod_{\underset{\gcd(k,n)=1}{k=1}}^n(X-e^{\frac{2ik\pi}{n}}).$$
I have shown already that $$X^n-1=\prod_{d\mid n}\Phi_d(X)$$ and that $$\Phi_p(X)=\prod_{k=1}^p(X-e^{\frac{2ik\pi}{p}})$$
but I'm not able to conclude that $\Phi_{p^n}(X)=\Phi_p(X^{p^{n-1}})$ with $p$ prime.