The dynamics of permutations on irreducible polynomials

TL;DR

This paper explores how permutations of irreducible polynomials over finite fields are induced by permutation polynomials, analyzing their dynamics and providing methods to generate such polynomials iteratively.

Contribution

It establishes that all permutations of irreducible polynomials of a fixed degree are induced by permutation polynomials over an extension field, and studies their dynamic properties.

Findings

Permutations of irreducible polynomials are induced by permutation polynomials.

Analyzed fixed points and cycle structures of these permutations.

Developed an iterative method to generate irreducible polynomials.

Abstract

We study degree preserving maps over the set of irreducible polynomials over a finite field. In particular, we show that every permutation of the set of irreducible polynomials of degree $k$ over $\mathbb{F}_q$ is induced by an action from a permutation polynomial of $\mathbb{F}_{q^k}$ with coefficients in $\mathbb{F}_q$. The dynamics of these permutations of irreducible polynomials of degree $k$ over $\mathbb{F}_q$, such as fixed points and cycle lengths, are studied. As an application, we also generate irreducible polynomials of the same degree by an iterative method.