Cyclotomic polynomial $\Phi_p$ for a prime $p$

For prime $p$, every nontrivial $p$th root is primitive, so $\Phi_p=1+x+\cdots+x^{p-1}$.Suggest a slogan

For a prime $p$, the cyclotomic polynomial $\Phi_p$ is \[ \Phi_p(x)=\frac{x^p-1}{x-1}=1+x+x^2+\cdots+x^{p-1}. \]

The roots of $x^p-1$ are the $p$th roots of unity, and every one of them other than $1$ is primitive because $p$ is prime. So $x^p-1=(x-1)\Phi_p(x)$, and dividing by $x-1$ with FF-ED3CD at $a=1$ gives the sum.