TITLE: How to prove that a complex number is not a root of unity?
QUESTION [30 upvotes]: $\frac35+i\frac45$ is not a root of unity though its absolute value is $1$.

Suppose I don't have a calculator to calculate out its argument then how do I prove it?
Is there any approach from abstract algebra or can it be done simply using complex numbers?
Any help will be truly appreciated.

REPLY [4 votes]: Let $a=\frac{3}{5}+\frac{4}{5}i$. If $a$ were a root of unity, there would exist a positive integer $n$ so that
$$a^n-1=0.$$
Expressions of the form $x^n-1$ can be factored into products of cyclotomic polynomials:
$$x^n-1=\prod_{d|n}\Phi_d(x).$$
There are several properties of cyclotomic polynomials that are relevant here:

$\Phi_d(x)\in\mathbb{Z}[x]$. That is, all its coefficients are integers.
$\Phi_d(x)$ is monic, meaning that the coefficient of the highest-powered term is 1.
$\Phi_d(x)$ is irreducible over $\mathbb{Q}[x]$. That is, it cannot be factored into two non-constant polynomials with rational coefficients.

The minimal monic polynomial $p(x)\in\mathbb{Q}[x]$ for $a$ is readily calculated to be
$$p(x)=x^2-\frac{6}{5}x+1.$$
In other words, $p(x)$ is the lowest-powered monic polynomial with rational coefficients where $p(a)=0$. Because of the non-integer $-\frac{6}{5}$ coefficient, $p(x)\neq\Phi_d(x)$ for any positive integer $d$ and therefore $a$ is not a root of unity.