Computing the Fixed Field of $\text{Aut}(\mathbb{F}(x)/\mathbb{F})$

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.