Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

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 $q$.