Irreducibility properties of Carlitz' binomial coefficients for algebraic function fields

Abstract

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