Alternatively, this is what's called the "norm map" $N_{\mathbb{Q}(\zeta)/\mathbb{Q}}(1-\zeta)$. Namely, what you are considering is
$\prod_{\sigma\in\text{Gal}(\mathbb{Q}(\zeta)/\mathbb{Q})}\sigma(1-\zeta)$
The cool thing it turns out is that $N_{\mathbb{Q}(\zeta)/\mathbb{Q}}(x)$ is just the determinant of the linear transformation determined by mulltiplication by $x$. In particular, we know that $1,\cdots,\zeta^{p-1}$ is a basis for $\mathbb{Q}(\zeta)/\mathbb{Q}$. Note then that if multiply by $1-\zeta$ we get the matrix
$\begin{pmatrix}1 & 0 & 0 & \cdots & 1\\ -1 & 1 & 0 & \cdots & 1\\ 0 & -1 & 1 & \cdots & 1\\ \vdots & \vdots & \vdots & \ddots & \vdots\\ 0 & 0 & \cdots & -1 & 2\end{pmatrix}$
with dimension $(p-1)\times(p-1)$--which has determinant $p$ as desired.