Let $f$ be an irreducible monical polynomial of in $\mathbb{F}_5[X]$ such that $\deg(f)=3$, and let $\alpha$ be a root in some field $\mathbb{F}_5^n$. Show that $31$ divides the order of $\alpha \in (\mathbb{F}_5^n)^*$.
My own work
The root $\alpha$ must belong to some field $\mathbb{F}_{125}$. There are $124 = 31 \cdot 2^2$ elements in $\mathbb{F}_{125}^*$. I know that $\alpha$ can't have the order $4$ or lower because the equality $x^5=x$ is a property specific to $\mathbb{F}_5$, and $\alpha$ does not belong to $\mathbb{F}_5$.
is this all right or are there mistakes in there. Please tell me.