## Abstract

We study the class of univariate polynomials β_{k}(X), introduced by Carlitz, with coefficients in the algebraic function field F_{q}(t) over the finite field F_{q} with q elements. It is implicit in the work of Carlitz that these polynomials form an F_{q}[t]-module basis of the ring Int(F_{q}[t])={f∈F_{q}(t)[X]|f(F_{q}[t])⊆F_{q}[t]} of integer-valued polynomials on the polynomial ring F_{q}[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=q^{s}, where s is a non-negative integer, that β_{k} is irreducible in Int(F_{q}[t]) and that it is even absolutely irreducible, that is, all of its powers β_{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 β_{k} is not even irreducible if k is not a power of q.

Original language | English |
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Article number | 102413 |

Journal | Finite fields and their applications |

Volume | 96 |

DOIs | |

Publication status | Published - Jun 2024 |

## Keywords

- Absolute irreducibility
- Carlitz polynomials
- Finite fields
- Function fields
- Integer-valued polynomials
- Irreducibility

## ASJC Scopus subject areas

- Theoretical Computer Science
- Algebra and Number Theory
- General Engineering
- Applied Mathematics