Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

TL;DR

This paper investigates the irreducibility of Carlitz's polynomials within the ring of integer-valued polynomials over finite fields, establishing their absolute irreducibility for powers of the characteristic.

Contribution

It proves that Carlitz's polynomials are irreducible and absolutely irreducible when their degree is a power of the finite field's size, extending understanding of their algebraic properties.

Findings

Carlitz's polynomials are irreducible for degrees that are powers of q.

These polynomials are absolutely irreducible, with unique factorization of their powers.

Irreducibility fails for degrees not a power of q.

Abstract

We study the class of univariate polynomials $\beta_k(X)$, introduced by Carlitz, with coefficients in the algebraic function field $\mathbb F_q(t)$ over the finite field $\mathbb F_q$ with $q$ elements. It is implicit in the work of Carlitz that these polynomials form a $\mathbb F_q[t]$-module basis of the ring $\text{Int}(\mathbb F_q[t]) = \{f \in \mathbb F_q(t)[X] \mid f(\mathbb F_q[t]) \subseteq \mathbb F_q[t]\}$ of integer-valued polynomials on the polynomial ring $\mathbb F_q[t]$. This stands in close analogy to the famous fact that a $\mathbb Z$-module basis of the ring $\text{Int}(\mathbb Z)$ is given by the binomial polynomials $\binom{X}{k}$. We prove, for $k = q^s$, where $s$ is a non-negative integer, that $\beta_k$ is irreducible in $\text{Int}(\mathbb F_q[t])$ and that it is even absolutely irreducible, that is, all of its powers $\beta_k^m$ with $m>0$ factor uniquely as products of irreducible elements of this ring. As we show, this result is optimal in the sense that $\beta_k$ is not even irreducible if $k$ is not a power of $q$.