Invariant Rational Functions, Linear Fractional Transformations and Irreducible Polynomials over Finite Fields

TL;DR

This paper explores how invariant rational functions under subgroup actions of PGL(2,q) can generate irreducible polynomials over finite fields, revealing regular factorization patterns and connections to group conjugacy classes.

Contribution

It introduces a novel method linking invariant rational functions to irreducible polynomial construction over finite fields, utilizing group actions and orbit polynomials.

Findings

Irreducible polynomials can be derived from invariant rational functions.

Regular patterns in polynomial factorization are identified.

Connections between PGL(2,q) conjugacy classes and polynomial properties are established.

Abstract

For a subgroup of $PGL(2,q)$ we show how some irreducible polynomials over $\mathbb{F}_q$ 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.