# Roots of unity

1. Jul 21, 2008

### 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:
$(1-w)(1-w^2)...(1-w^{n-1})=n$

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 $|1-w|^2 |1-w^2|^2...$
but I'm not sure what to do from here.
Thanks

Last edited by a moderator: Jul 21, 2008
2. Jul 21, 2008

### HallsofIvy

Staff Emeritus
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, ..., wn-1 and so xn-1= (x-1)(x-w)(x-w2)...(x- wn-1). Dividing both sides by x- 1 we get (x-w)(x-w2)...(x- wn-1) on the right and what on the left? Now set x= 1.

3. Jul 21, 2008

thanks