Proving Primitive Root of Unity: z = cos(2pi/n) + isin(2pi/n)

[Driessen12]

Homework Statement

show that cos(2pi/n) + isin(2pi/n) is a primitive root of unity

Homework Equations

The Attempt at a Solution

if i know z = cos(2pi/n) + isin(2pi/n) is an nth root and I'm trying to prove that z is a primitive nth root. is it correct to assume that z^k is not primitive and is therefore an nth root of unity. I just need to know if z is an nth root and i know z^k is not primitive, does it have to be an nth root? just want to make sure there are no subtleties. I have the rest of the proof. I think this is easier if I use contradiction and that would require me to assume that if z^k is not primitive that it is an nth root.

 

[Landau]

I think it will be quite helpful (to you) to state the precise definitions of (n-th) root and primitive (n-th) root of unity. Also Moivre[/url] can be useful here.