TITLE: Not a root of unity - how to prove?
QUESTION [6 upvotes]: 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 $2^n = \frac{1}{e^{i \pi n/2}}$?

REPLY [2 votes]: 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.