Proof about Constructibility of complex numbers

[AlexChandler]

Homework Statement

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

then $$p=2^k+1$$ for a positive integer k

Homework Equations

$$e^{i \theta} = Cos \theta + iSin \theta$$

The Attempt at a Solution

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

$$Cos(2 \pi /p) , Sin(2 \pi /p)$$ 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

 

[morphism]

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?