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 $\text{ P}\Gamma \text{ L}(2, q^n)=\text{ PGL}(2, q^n)\rtimes \text{ Gal}(\mathbb{ F}_{ q^n}/\mathbb{ F}_q)$ 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.
Categories: math.NT