We have a vanishing polynomial $Z(X) = X^N - 1$. Implicitly we are proving that

$$X^N = 1$$

What are the solutions to this polynomial? Well the answer is $\omega$ which is any root of $1$.

Therefore the solution to the formula $X^N - 1$ will be all the values of $X^N = 1$ or

$$X^N - 1 = (\omega - 1)(\omega^2 - 1)\cdots(\omega^{N - 1} - 1)$$