Why isnt product of nth roots of unity always 1 I know product of nth roots of unity is 1 or -1 depending whether n is odd or even.
But in this way I am getting 1. Where am I wrong?
$ \text{Let }\alpha = \cos \frac{2 \pi}{n} + \iota \sin \frac{2 \pi}{n} \text{ be a root of }x^n=1 \\
\text{Then product of nth roots will be } 1\cdot \alpha \cdot \alpha^2 ... \alpha^{n-1} = \alpha^{\frac{(n)(n-1)}{2}}\\
=\left( \alpha^n \right)^{\frac{n-1}{2}}\\
=1^{\frac{n-1}{2}} \text{......By the definition of alpha ??}\\
=1 $
I can see this doesn't even work for n=2.

Hint:
$$\alpha^{n(n-1)/2}=(\alpha^{n/2})^{n-1}$$
Now $\alpha^{n/2}=\cdots=-1$
