What can we say about the dimension of $[ E : F]$ if $F$ is finite and $f \in F[x]$ min. polynomial

Question

suppose I have a field $F$ and $\alpha \notin F$. $F$ is finite so $char(F)$ is some $p \in \mathbb{N}$

when I have a minimal polynomial $f_\alpha \in F[x]$ with $deg(f_\alpha)=n$ then the dimension of $[F(\alpha) : F ] = n$ is there also some relation to the $char(F)$ ?

Answer 1

It is known that if $F$ is a finite field, then for each integer $n$ there exists an irreducible polynomial of degree $n$ over $F$. So we cannot have any connection between the characteristic of $F$ and $n$.