roots of unity

plmokn2

1. The problem statement, all variables and given/known data
if w is the nth root of unity, i.e. w= exp(2pi/n i) show:
[latex](1-w)(1-w^2)...(1-w^{n-1})=n[/latex]


2. Relevant equations



3. The attempt at a solution
since w^(n-a)= complex congugate of w^a
terms on the left hand side are going to pair up to give [latex]|1-w|^2 |1-w^2|^2...[/latex]
but I'm not sure what to do from here.
Thanks

HallsofIvy

I wouldn't do it that way at all!

It should be sufficient to note that the n roots of xⁿ- 1= 0 are 1, w, w², ..., w^n-1 and so xⁿ-1= (x-1)(x-w)(x-w²)...(x- w^n-1). Dividing both sides by x- 1 we get (x-w)(x-w²)...(x- w^n-1) on the right and what on the left? Now set x= 1.

plmokn2

thanks