Extension fields and integer-valued polynomials
type: Item
instance of: scholarly article
title: Extension fields and integer-valued polynomials
zbMATH Open document ID: 0920.13010
author: Catalin Barbacioru
published in: Journal of Number Theory
publication date: 1999-09-02T00:00:00Z
review text: Let \(A\) be a Dedekind domain with finite residue fields and quotient field \(K\), and let \(\text{Int}(A)=\{f\in K[X]\mid f(A)\subset A\}\) be the ring of integer valued polynomials for \(A\). If \(L\) is a finite separable extension of \(K\) and \(B\) is the integral closure of \(A\) in \(L\), one may form \(\text{Int}(B)\) and ask how it is related to \(\text{Int}(A)\). The paper under review shows that \(\text{Int}(A)\subseteq \text{Int}(B)\) if and only if all maximal ideals of \(A\) split completely in \(L\) and that in this case \(\text{Int}(B)\) is the integral closure of \(\text{Int}(A)\) in \(L(X)\). The proof proceeds by studying essential valuations on \(K(X)\) (i.e. valuations whose valuation ring is the localization of \(\text{Int}(A)\) at a prime ideal) and analyzing the extensions of these to \(L(X)\).
reviewed by: Keith Johnson
Mathematics Subject Classification ID: 13F20
Mathematics Subject Classification ID: 13B22
Mathematics Subject Classification ID: 11R09
Mathematics Subject Classification ID: 13A18
Mathematics Subject Classification ID: 12F10
zbMATH DE Number: 1268042
zbMATH Keywords: ring of integer valued polynomials
zbMATH Keywords: integral closure
zbMATH Keywords: essential valuations
MaRDI profile type: Publication
full work available at URL: jnth.1998.2267
OpenAlex ID: W2066078441
cites work: Q5625315
cites work: Anneaux de 'polynômes à valeurs entières' et anneaux de Fatou
cites work: Q5661989
cites work: Split primes and integer-valued polynomials
DOI: 10.1006/JNTH.1998.2267
description: scientific article; zbMATH DE number 1268042