Invariant theory of a special group action on irreducible polynomials over finite fields

TL;DR

This paper studies the action of the projective linear group on irreducible polynomials over finite fields, providing comprehensive theoretical results on invariant elements and their enumeration.

Contribution

It generalizes previous work on the group action, offering complete characterizations and counts of invariant irreducible polynomials over finite fields.

Findings

Characterization of invariant irreducible polynomials

Exact formulas for counting invariant elements

Generalization of previous partial results

Abstract

In the past few years, an action of $\mathrm{PGL}_2(\mathbb F_q)$ on the set of irreducible polynomials in $\mathbb F_q[x]$ has been introduced and many questions have been discussed, such as the characterization and number of invariant elements. In this paper, we analyze some recent works on this action and provide full generalizations of them, yielding final theoretical results on the characterization and number of invariant elements.