Integral-valued polynomials over the set of algebraic integers of bounded degree

Abstract

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $h\in K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)\subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $f\in\mathbb{Q}[X]$: if $f(\alpha)$ is integral over $\mathbb{Z}$ for every algebraic integer $\alpha$ of degree $n$, then $f(\beta)$ is integral over $\mathbb{Z}$ for every algebraic integer $\beta$ of degree smaller than $n$. This second result is established by proving that the integral closure of the ring of polynomials in $\mathbb{Q}[X]$ which are integer-valued over the set of matrices $M_n(\mathbb{Z})$ is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to $n$.