  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}.
