1. PF Contest - Win "Conquering the Physics GRE" book! Click Here to Enter
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

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


    User Avatar
    Science Advisor
    Homework Helper

    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?
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Threads - Proof Constructibility complex Date
Lim sup proof Mar 12, 2018
Real Analysis Proof Mar 11, 2018
Proof that a recursive sequence converges Mar 8, 2018
Bland rule proof linear programming Mar 7, 2018
Please help construct a proof (propositional logic) Dec 3, 2011