In my algebra course we just started the topic of Permutation Polynomial, and I am trying to prove that is $f \in F_q[x]$ is a permutation polynomial over $F_{q^n}$, $n \geq 1$, then $f$ is a permutation polynomial on any subfield of $F_{q^n}$ that contains $F_q$.
I know that $F_{q^m}$ is a subfield of $F_{q^n}$ containing $F_{q}$ if and only if $m|n$, so I am trying to argue that the degree of the permutation polynomial $f$ and $n$ must be coprime, but I am not sure of how to develop this argument (I am also not completely sure is right).
Any hint?
Thank you.
Note that $f\in\mathbb{F}_q[x]\subseteq\mathbb{F}_{q^m}[x]$, where $\mathbb{F}_{q^m}$ is a subfield of $\mathbb{F}_{q^n}$ containing $\mathbb{F}_q$. Since $f$ permutes $\mathbb{F}_{q^n}$, $f$ is injective, and maps $\mathbb{F}_{q^m}$ to itself. An injection on a finite set is obviously a permutation.