Let $a$ be an integer such that $a$ is not the $d$-th power of an integer, for $d$ a divisor of $n$, $a>1$ and let $\alpha$ the unique positive real $n-th$ root of $a$. Supppose that $(m,n)=d$, for $d$ a divisor of $n$. Let $L$ be
$$L=\prod_{j=1}^{n}(1-\alpha^{m}\zeta_{n}^{jm})$$
I must prove that
$$L=(-1)^n \left(\prod_{k|\frac{n}{d}}\phi_{k}(\alpha^{m/d})\right)^d$$
My Aproach
$$L=\prod_{j=1}^{n}(1-\alpha^{m}\zeta_{n}^{jm})$$ $$=\prod_{j=1}^{n}(\zeta_{n}^{jm}-\alpha^{m})$$
We assume $\gcd(m,n)=d$, so $\gcd(m/d,n/d)=1$. Therefore $\zeta_{n}^{m}=\zeta_{n/d}^{m/d}$. It follows that
$$L=(-1)^{n}\prod_{j=1}^{n}(\alpha^{m}-\zeta_{n/d}^{jm/d})$$
and then I can not complete the proof. Does anybody has a hint?