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

TL;DR

This paper constructs an explicit, easily computable element in the rational function field over finite fields that generates the fixed field of automorphisms fixing the base field, making Galois theory more accessible.

Contribution

It provides an elementary, explicit construction of a generator for the fixed field of automorphisms over finite fields, simplifying Galois theory applications.

Findings

Explicit formula for the generator of the fixed field

Elementary approach suitable for students

Applicable to all finite fields

Abstract

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.