Showing the Equivalence of (1-w)(1-w^2)...(1-w^{n-1}) and n

[plmokn2]

Homework Statement

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

Homework Equations

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

 

[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