Product related to $n^{\rm th}$ roots of unity.

Question

How to find $$\prod_{i=1}^{n-1}(1+\alpha_i)\quad\alpha_i\text{ are the roots of }z^{n}=1$$

Answer 1

You're looking at the product of the roots of the polynomial $(z-1)^n - 1$. But, the product of the roots of a monic polynomial is the constant term multiplied by $(-1)^n$. Therefor, the product is equal to $0$ if $n$ is even, and equal to $2$ if $n$ is odd.