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Proof about Constructibility of complex numbers

  1. Jan 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Show that if p is prime and [tex] e^{2 \pi i/p} [/tex] is constructable

    then [tex] p=2^k+1 [/tex] for a positive integer k


    2. Relevant equations

    [tex] e^{i \theta} = Cos \theta + iSin \theta [/tex]


    3. The attempt at a solution

    By definition, a complex number a+bi is constructible if a and b are constructible. Thus we know that

    [tex] Cos(2 \pi /p) , Sin(2 \pi /p) [/tex] are constructible

    I have tried finding a polynomial such that these are roots but I am having trouble here. We have a theorem that if a real number is a root of a polynomial of some degree that is not a power of 2, then the number is not constructible. I am trying to use this to show that p must be one more than a power of 2, but i'm not sure how to construct these polynomials. Any ideas? Thanks
     
    Last edited: Jan 22, 2012
  2. jcsd
  3. Jan 28, 2012 #2

    morphism

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    If you know that cos(2pi/p) and sin(2pi/p) are constructible, then you can conclude that the degree of the extension Q(cos(2pi/p),sin(2pi/p))/Q is a power of two (why?). What does this tell you about the degree of Q(e^{2pi i/p})/Q?
     
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