Let $q = p^r$ some prime power.
Let $f \in \mathbb{F}_{q^m}$, some field extension of $\mathbb{F}_{q}$. Then, if $f = f^q$, then $x \in \mathbb{F}_q$.
Why is this true? Is it simply because $x$ is a root of the polynomial $x^q - x \in \mathbb{F}_q[x]$ and $\mathbb{F}_q[x]$ consists precisely of the roots of this polynomial?