In Washington's Cyclotomic Fields, he makes the following assertion (p. 233): For each positive integer $i$, let $\zeta_i$ be a primitive $i$-th root of unity, chosen such that $(\zeta_i)^{(i/j)}=\zeta_j$. Then
$\prod_{b=0}^{(i/j)-1} (\zeta_i^{a+bj}-1)=\zeta_j^a-1$ if $j | i$.
After playing around with this for a while and working a few small examples, I still can't see how to prove this generally, even though I'm sure it's very simple. Can someone please enlighten me? Thanks!