On $2$-superirreducible polynomials over finite fields

TL;DR

This paper studies the existence and enumeration of 2-superirreducible polynomials over finite fields, revealing nonexistence in certain cases and providing explicit formulas and asymptotic analysis for others.

Contribution

It establishes nonexistence results for 2-superirreducible polynomials in specific cases and derives an explicit counting formula for even degree polynomials over finite fields.

Findings

No 2-superirreducible polynomials exist over fields of characteristic 2.

No such polynomials of odd degree exist when the characteristic is odd.

An explicit formula for counting 2-superirreducible polynomials of even degree is provided.

Abstract

We investigate $k$-superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$. Let $\mathbb F$ be a finite field of characteristic $p$. We show that no $2$-superirreducible polynomials exist in $\mathbb F[t]$ when $p=2$ and that no such polynomials of odd degree exist when $p$ is odd. We address the remaining case in which $p$ is odd and the polynomials have even degree by giving an explicit formula for the number of monic 2-superirreducible polynomials having even degree $d$. This formula is analogous to that given by Gauss for the number of monic irreducible polynomials of given degree over a finite field. We discuss the associated asymptotic behaviour when either the degree of the polynomial or the size of the finite field tends to infinity.