  For a finite field $\mathbb{F}$, it is a basic result of Galois theory that
the fixed field $E$ of $\text{Aut}(\mathbb{F}(x)/\mathbb{F})$ is a proper
extension of $\mathbb{F}$. In this expository paper we construct, for all
finite fields, an element $f\in \mathbb{F}(x)$ such that $E = \mathbb{F}(f)$.
This element is shown to be easily computable in all cases using a simple
formula. The approach is entirely elementary, and aimed at the student level;
only a basic familiarity with Galois theory and field extensions is assumed.
