I stumbled upon one of these exercises in my textbook and thought it would be a great review for my exam. Here is the question:
Let $n$ be an odd positive integer, and let $\zeta$ be a primitive (complex) $n$-th root of unity. Show that $\zeta^2$ is also a primitive $n$-th root of unity.
This is what I know:
$\zeta$ is a primitive $n$-th root of unity where $O(\zeta) = n$. So in this case,I began to list some of the odd positive integers, where $$n = {1,3,5,7,9,...}$$
Then I considered the number $\sqrt[n]{1}$, where I can derive the appropriate roots based on the value of $n$ I choose. So as an example, $\sqrt[3]{1}$ = {$1,\omega, \omega^2$},
where $\omega$ = $-\frac{1}{2} + \frac {\sqrt{3}}2i$ and $\omega^2$ = $-\frac{1}{2} - \frac {\sqrt{3}}2i$. Is this even the right approach? I want to move on to other problems but not until I figure out what's going on here.
All primitive $n$th roots of unity have the form $\zeta=\exp(\frac{2\pi ik}{n})$ with $k$ an integer such that $(k,n)=1$.
So it is enough to observe that $\zeta^2=\exp(\frac{2\pi i\cdot 2k}{n})$ and that if $k$ is prime to $n$ then so is $2k$, since $n$ is odd.