Let $\zeta_p$ be a $p$-th root of unity, where $p$ is an odd prime number.
I just came across the following expression:
$$\frac{(\zeta_p^2-\zeta_p+1)^3}{\zeta_p^2(\zeta_p-1)^2}.$$ Can we simplify this expression somehow? For $p=3$, I can rewrite this to $$\frac{8}{(\zeta_3-1)^2}.$$
Question. Can we simplify $$\frac{(\zeta_p^2-\zeta_p+1)^3}{\zeta_p^2(\zeta_p-1)^2} ?$$
Note that this expression does not change if we replace $\zeta_p$ by one of the elements in $\{\zeta_p, \zeta_p^{-1},1-\zeta_p, (1-\zeta_p)^{-1}, \zeta_p/(1-\zeta_p), (1-\zeta_p)/\zeta_p\}$.