$p=(1-\zeta_{p})...(1-\zeta^{p-1}_{p})$ Let $\zeta_{p}$ be a primitive $p$th root of $1$. Then $t^{p}-1=(t-1)(t-\zeta_{p})(t-\zeta_{p}^{2})\ldots(t-\zeta_{p}^{p-1})$. Using this, I need to show that $p=(1-\zeta_{p})\ldots(1-\zeta^{p-1}_{p})$ but I do not know how.

$$(t-\zeta_{p})(t-\zeta_{p}^{2})\ldots(t-\zeta_{p}^{p-1})=\frac{t^p-1}{t-1}=1+t+t^2+...+t^{p-1}$$
plug in $t=1$.
