The action of $\rm{GL}_2(\mathbb{F}_q)$ on irreducible polynomials over $\F_q$

TL;DR

This paper investigates the fixed irreducible polynomials under the action of subgroups of rac{GL}_2(\u00Fq) over finite fields, providing complete characterizations for specific subgroup types and alternative solutions for others.

Contribution

It offers a complete characterization of fixed irreducible polynomials under certain subgroup actions of rac{GL}_2(\u00Fq), extending previous partial results.

Findings

Characterization of fixed polynomials for translation subgroups

Results for p-subgroups of rac{GL}_2(_q)

Alternative solutions for diagonal and rac{PGL}_2(_q) cases

Abstract

Let $\F_q$ be the finite field with $q$ elements, $p=\Char \F_q$. The group $\GL_2(\F_q)$ acts naturally in the set of irreducible polynomials over $\F_q$ of degree at least $2$. In this paper we are interested in the characterization and number of the irreducible polynomials that are fixed by the elements of a subgroup $H$ of $\GL_2(\F_q)$. We make a complete characterization of the fixed polynomials in the case when $H$ has only elements of the form $\left(\begin{matrix} 1&b\\ 0&1 \end{matrix}\right)$, corresponding to translations $x\mapsto x+b$ and, as a consequence, the case when $H$ is a $p-$subgroup of $\GL_2(\F_q)$. This paper also contains alternative solutions for the cases when $H$ is generated by an element of the form $\left(\begin{matrix} a&0\\ 0&1 \end{matrix}\right)$, obtained by Garefalakis (2010) and $H=\rm{PGL}$$_2(\F_q)$, obtained by Stichtenoth and Topuzoglu (2011).