Let $\alpha>1$, $p$ a prime number and $\phi_{k}(X)$ the $k$-th cyclotomic polynomial. I want to prove that
$$ \phi_{k}(\alpha^p)-\phi_{k}(\alpha)\neq 0 $$
My attempt
Still I do not have any aproach because the generality of the problem. However, since $\alpha>1$, I think that $\alpha^p$ "must to dominate"
Hint: Cyclotomic polynomials are products of linear factors of the form $X-\zeta$, where $\zeta$ is on the unit circle. If $\alpha,p>1$ then $\alpha^p$ is further from the unit circle than $\alpha$.