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Roots of unity

  1. Jul 21, 2008 #1
    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:
    [itex](1-w)(1-w^2)...(1-w^{n-1})=n[/itex]


    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 [itex]|1-w|^2 |1-w^2|^2...[/itex]
    but I'm not sure what to do from here.
    Thanks
     
    Last edited by a moderator: Jul 21, 2008
  2. jcsd
  3. Jul 21, 2008 #2

    HallsofIvy

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    Staff Emeritus
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    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.
     
  4. Jul 21, 2008 #3
    thanks
     
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