# Explicit Artin maps into ${\rm PGL}_2$

## TL;DR

This paper explicitly computes the Artin map for subgroups of PGL_2 over finite fields, revealing new insights into finite fields, additive polynomials, and conjugacy classes, with applications to characterizations and symbols.

## Contribution

It provides explicit calculations of the Artin map for various subgroups of PGL_2 over finite fields, leading to new characterizations and structures related to finite fields and polynomials.

## Key findings

- New characterization for when additive polynomials split over finite fields
- Information about conjugacy classes of PGL_2(F_q)
- A natural tripartite symbol on F_q with values in Z/3Z

## Abstract

Let $G$ be a subgroup of ${\rm PGL}_2({\mathbb F}_q)$, where $q$ is any prime power, and let $Q \in {\mathbb F}_q[x]$ such that ${\mathbb F}_q(x)/{\mathbb F}_q(Q(x))$ is a Galois extension with group $G$. By explicitly computing the Artin map on unramified degree-1 primes in ${\mathbb F}_q(Q)$ for various groups $G$, interesting new results emerge about finite fields, additive polynomials, and conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. For example, by taking $G$ to be a unipotent group, one obtains a new characterization for when an additive polynomial splits completely over ${\mathbb F}_q$. When $G = {\rm PGL}_2({\mathbb F}_q)$, one obtains information about conjugacy classes of ${\rm PGL}_2({\mathbb F}_q)$. When $G$ is the group of order 3 generated by $x \mapsto 1 - 1/x$, one obtains a natural tripartite symbol on ${\mathbb F}_q$ with values in ${\mathbb Z}/3{\mathbb Z}$. Some of these results generalize to ${\rm PGL}_2(K)$ for arbitrary fields $K$. Apart from the introduction, this article is written from first principles, with the aim to be accessible to graduate students or advanced undergraduates. An earlier draft of this article was published on the Math arXiv in June 2019 under the title {\it More structure theorems for finite fields}.