$p$ is a prime, and $\Phi_{p-1}(x)$ denote the cyclotomic polynomial of order $p-1$. And I want to show the following:
$g$ is a solution of the congruence $\Phi_{p-1}(x) \equiv 0 (mod~p)$ if and only if $g$ is a primitive root (mod p)
that is:
$\Phi_{p-1}(g) \equiv 0 (mod~p) \iff g$ is a primitive root (mod p)
Here is some properties about cyclotomic polynomial:
${\textstyle \prod_{d|n}^{}}\Phi_{d}(x) = x^n-1\tag{1}$
$\Phi_{n}(x) = {\textstyle \prod_{d|n}^{}}(x^d-1)^{\mu(n/d)}\tag{2}$
$\mu(x)$ is the Möbius inversion formula