Let $G\subset {\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$ where $q$ is any prime power, and let $Q\in {\mathbb{F}}_q\left(x\right)$ such that ${\mathbb{F}}_q\left(x\right)/{\mathbb{F}}_q\left(Q\right)$ is a Galois extension with group $G$. By explicitly computing the Artin map on unramified degree-1 primes in ${\mathbb{F}}_q\left(Q\right)$ for various groups $G$, interesting new results emerge about finite fields, additive polynomials, and conjugacy classes of ${\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$. 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 $G={\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$, one obtains information about conjugacy classes of ${\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$. When $G$ is the group of order 3 generated by $\left(\genfrac{}{}{0ex}{}{1}{1}\genfrac{}{}{0ex}{}{-1}{0}\right)$, one obtains a natural tripartite symbol on ${\mathbb{F}}_q$ with values in $\mathbb{Z}/3\mathbb{Z}$. Some of these results generalize to ${\mathrm{PGL}}_2\left(K\right)$ for arbitrary fields $K$.

让$G\subset {\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$在哪里$q$是任何素幂，并且让$问\in {\mathbb{F}}_q\left(X\right)$这样${\mathbb{F}}_q\left(X\right)/{\mathbb{F}}_q\left(问\right)$是一个带群的伽罗瓦扩展$G$. 通过在未分支的 1 次素数上显式计算 Artin 映射${\mathbb{F}}_q\left(问\right)$针对不同的群体$G$，关于有限域、加法多项式和共轭类的有趣的新结果出现了${\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$. 例如，通过采取$G$作为一个单能群，当一个加法多项式完全分裂时，可以获得一个新的表征${\mathbb{F}}_q$. 什么时候$G={\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$, 获得关于共轭类的信息${\mathrm{PGL}}_2\left({\mathbb{F}}_q\right)$. 什么时候$G$是由生成的 3 阶组$\left(\genfrac{}{}{0ex}{}{1}{1}\genfrac{}{}{0ex}{}{-1}{0}\right)$, 得到一个自然的三方符号${\mathbb{F}}_q$与值$\mathbb{Z}/3\mathbb{Z}$. 其中一些结果概括为${\mathrm{PGL}}_2\left(ķ\right)$对于任意字段$ķ$.