If $\varepsilon$ is a $n$-th root of unity, then it is a $d$-th primitive root of unity for some $d | n$, in a special case.

Question

Let $F$ be a field, with $p = Char F$ a natural prime number and let $n \in \mathbb{N}$ such that $p | n$. Let $\varepsilon$ be a root of $X^n - 1$. How can I see that there exists $d \in \mathbb{N}$, with $d | n$, such that $\varepsilon$ generates the group of $d$-th roots of unity?