This is the first exercise in section 6.5 of Robert Ash's abstract algebra, I want to understand the given solution:
Noting that $\psi_n(X^p)= \prod_{w_i} X^p- w_i$, the roots of $X^p −ω_i$ are the $p$-th roots of $ω_i$, which must be primitive $np$-th roots of unity because the map $\theta\to \theta^p$ is a bijection between primitive $np$-th roots of unity and primitive $n$-th roots of unity ( because $\phi(np) = p\phi(n)$ as $p|n$ ).
I am assuming here $\phi$ is Euler's function (we can show the above equality by induction) but what is the link with that bijection?
