2021/05/24 by Rod Gow, Gow, Rod, Gary McGuire +1
Mathematics · #FOS: Mathematics #Number Theory (math.NT) #math.NT
paper · pdf · doi:10.48550/arxiv.2105.11247
New section added
arxiv created 2021/08/26 · arxiv updated 2021/08/27
For a subgroup of PGL(2,q) we show how some irreducible polynomials over \mathbbFq arise from the field of invariant rational functions. The proofs rely on two actions of PGL(2,F), one on the projective line over a field F and the other on the rational function field F(x). The invariant functions in F(x) are used to show that regular patterns exist in the factorization of certain polynomials into irreducible polynomials. We use some results about group actions and the orbit polynomial, whose proofs are included. An unusual connection to the conjugacy classes of PGL(2,q) is shown. At the end of the paper we present an alternative approach, using Lang's theorem on algebraic groups.