Roots of unity product

Question

For each $n \in \mathbb N, n \geq 3$ calculate the product of all the n roots of unity.

Or to say it in a more stric way:

$$\prod_{w \in G_n^*}w$$

Being $G_n^*$ the primitive roots of the unity.

Answer 1

Hint If $w$ is a root of unity, so is $\frac{1}{w}$.

So, as long as $w \neq \frac{1}{w}$ they can be used to cancel each other. The only thing what you need to worry about is:

What if $w=\frac{1}{w}$ for some $w$? Those roots are left by themselves. Note that $w=\frac{1}{w}$ has 2 roots, one doesn't matter and the other is a root of unity only for certain n's.