I know there are questions talking about similar issues here, but I don't understand a step that in the answers seem obvious steps.
Given a finite field $F_q$,
when I have a primitive n-th root of unity $\eta$ over $F_q$, I know that $\eta \in F_{q^d}$ (where $d = O_n(q)$). ($*$)
If $P(X)$ is the minimal polynomial of $\eta$ in $F_q[x]$, is $\eta \in F_q[x]/P(X)$?
I'm trying to understand why $(*)$ implies that the minimal polynomial of $\eta$ in $F_q[x]$ has degree $d$.