# Primitive nth roots of unity related to the complex nth roots of 1.

I just cannot seem to wrap my head around this problem and would really appreciate some guidance. So I know that a primitive $$n^{th}$$ root of unity is a complex number $$z$$ such that $$z^n = 1$$ but $$z^m \neq 1$$ for $$0. I also know that for $$w = cos(\frac{2\pi}{n}) + isin(\frac{2\pi}{n})$$, the $$n$$ complex solutions of $$z^n = 1$$ are $$1,w,w^2,...,w^{n-1}$$. I need to show three things:

1.) $$w$$ is always a primitive $$n^{th}$$ root of unity.
2.) $$w^k$$ is a primitive $$n^{th}$$ root of unity iff $$gcd(k,n) = 1$$
3.) If $$z$$ is any primitive $$n^{th}$$ root of unity, then $$1,z,z^2,...,z^{n-1}$$ are distinct and comprise all the $$n^{th}$$ roots of unity.

I'm pretty new to this type of rigorous proof and I'm not even sure where to start.

• So you have to show to start that $w^n=1$ and $w^k\ne 1$ if $0<k<n$. – xarles Oct 3 '18 at 19:12