In a proof in my syllabus of a number theory course, they use that
$(\mathbb{F}_q[x]/(f_i(x)))^{F_q} = \mathbb{F}_q$
where $f_i(x)$ is irreducible, $F_q$ is the frobenius automorphism of $\mathbb{F}_q$ and $\mathbb{F}_q$ is the finite field of size $q$ (not necessarily prime). The book also says that this can be proven using properties of finite fields. However, I flipped through me algebra book and I didn't see any particular property that I could use to show this.
Could anyone give me some pointers?