Möbius-Frobenius maps on irreducible polynomials

Abstract

Let $n$ be a positive integer and let ${\mathbb{F}}_{q^n}$ be the finite field with $q^n$ elements, where $q$ is a power of a prime. This paper introduces a natural action of the Projective Semilinear Group $\text{P}\Gamma \text{L}(2,q^n)=\text{PGL}(2,q^n)⋊\text{Gal}({\mathbb{F}}_{q^n}/{\mathbb{F}}_q)$ on the set of monic irreducible polynomials over the finite field ${\mathbb{F}}_{q^n}$. Our main results provide information on the characterization and number of fixed points.

12 pages; comments welcome