Let $Q_r(X)$ be the $r^{th}$ cyclotomic polynomial over $F_p$. Polynomial $Q_r(X)$ divides $X^r-1$ and factors into irreducible factors of degree $o_r(p)$. Let $h(X)$ be one such irreducible factor.
In the paper (Prime is in P), the authors say that "since $h(X)$ is a factor of the cyclotomic polynomial $Q_r(X)$, then $X$ is a primitive $r^{th}$ root of unity in $F=F_p[X]/h(X)$".
For me it's easy to see that it is a $r^{th}$ root of unity, but I can't see why it is a primitive $r^{th}$ root of unity. I can't get why the order of $X$ is exactly $r$ in $F$.
Edit: p is prime