Commutative Ring Theory

5 Polynomials Whose Derivatives Are Integer-Valued in Number Fields

JEAN-LUC CHABERT Institut Supérieur des Sciences et Techniques. Université de Picardie. 48 rue Raspail, 02109 St Quentin, France

ABSTRACT: Carlitz proved that polynomials whose derivatives are integer-valued on Z are also those whose divided differences are integer-valued. We show that this result holds for the ring of integers A of some number fields. More precisely it holds if and only if each maximal ideal of A lying over a prime number p has an absolute ramification index strictly less than p.

1. Introduction

Carlitz's theorem

Everybody knows that:

1.1. The set B of integer-valued polynomials on $\mathbb{Z}$, i.e. ${\displaystyle B}=\{\text{P}\in \mathbb{Q}[\text{X}]1\text{P}(\mathbb{Z})\subset \mathbb{Z}\}$, is a free $\mathbb{Z}\text{−module}$ generated by the binomial ...