Invariant rational functions, linear fractional transformations and irreducible polynomials over finite fields

For a subgroup of PGL(2, q) we show how some irreducible polynomials over F-q arise from the field of invariant rational functions. The proofs rely on combining 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. (C) 2021 Elsevier Inc. All rights reserved.