I'm working on a problem, but I haven't been able to get much out of it...
Given that $\zeta$ is a primitive complex n-th root of unity, show that the complex numbers $1,\zeta,\zeta^2,...,\zeta^{n-1}$ are all distinct, meaning that no two of them are equal. Note that it is not specified which primitive n-th root $\zeta$ is.
I believe I have to incorporate the orders of these $\zeta$ elements in some way. From what I know, a primitive complex n-th root of unity is when $O(\zeta) = n$, where $O(\zeta)$ is the order of $\zeta$. With this in mind, how would I show each of the elements, $1,\zeta,\zeta^2,...,\zeta^{n-1}$ are all distinct..?