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 $n$.

keywords: Integer-valued polynomial, Algebraic integers with bounded degree, Prüfer domain, Polynomially dense subset, Integral closure. To appear in Journal of Number Theory

References in corpus (1)

• Integer-valued polynomials on algebras

Cited by in corpus (8)

• Properly Integral Polynomials over the Ring of Integer-valued Polynomials on a Matrix Ring
• Integral closure of rings of integer-valued polynomials on algebras
• The ring of polynomials integral-valued over a finite set of integral elements
• Non-triviality Conditions for Integer-valued Polynomial Rings on Algebras
• Galois structure on integral valued polynomials
• Decomposition of integer-valued polynomial algebras
• A Survey on Fixed Divisors
• Nontriviality of rings of integral-valued polynomials