Not a root of unity - how to prove?

Question

How does one prove that a given complex number is not a root of unity for some positive integer power n? Say, I want to prove that there does not exist a positive integer $n$ such that $(2i)^n = 1$, I could argue that $(2i)^n = 2^ne^{i \pi n/2}$, and we know that $2^n \neq 1$ for any positive integer $n$. But how can we prove that there is no positive integer $n$ such that $2^n = \frac{1}{e^{i \pi n/2}}$?

Answer 1

For any real $a$, $x =e^{i\pi a} =\cos(\pi a)+i\sin(\pi a) $ so $|x| = 1$.

Therefore, since $|uv| = |u| |v|$ for any complex $u$ and $v$, then, for any complex $c$ and real $a$, $|c e^{i\pi a}| =|c| |e^{i\pi a}| =|c| $.

Therefore, if $|c| \ne 1$, $c e^{i\pi a} $ can not be a root of unity.