Explicit Artin maps into [formula omitted

Abstract

Let G⊂PGL2(Fq) where q is any prime power, and let Q∈Fq(x) such that Fq(x)/Fq(Q) is a Galois extension with group G. By explicitly computing the Artin map on unramified degree-1 primes in Fq(Q) for various groups G, interesting new results emerge about finite fields, additive polynomials, and conjugacy classes of PGL2(Fq). For example, by taking G to be a unipotent group, one obtains a new characterization for when an additive polynomial splits completely over Fq. When G=PGL2(Fq), one obtains information about conjugacy classes of PGL2(Fq). When G is the group of order 3 generated by 11−10, one obtains a natural tripartite symbol on Fq with values in Z/3Z. Some of these results generalize to PGL2(K) for arbitrary fields K.