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Integral-valued polynomials over sets of algebraic integers of bounded degree()

Let K be a number field of degree n with ring of integers [Formula: see text]. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if [Formula: see text] maps every element of [Formula: see text] of degree n to an algebraic intege...

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Autor principal: Peruginelli, Giulio
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Elsevier 2014
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4750901/
https://www.ncbi.nlm.nih.gov/pubmed/26949270
http://dx.doi.org/10.1016/j.jnt.2013.11.007
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author Peruginelli, Giulio
author_facet Peruginelli, Giulio
author_sort Peruginelli, Giulio
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description Let K be a number field of degree n with ring of integers [Formula: see text]. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if [Formula: see text] maps every element of [Formula: see text] of degree n to an algebraic integer, then [Formula: see text] is integral-valued over [Formula: see text] , that is, [Formula: see text]. A similar property holds if we consider the set of all algebraic integers of degree n and a polynomial [Formula: see text]: if [Formula: see text] is integral over [Formula: see text] for every algebraic integer α of degree n, then [Formula: see text] is integral over [Formula: see text] for every algebraic integer β of degree smaller than n. This second result is established by proving that the integral closure of the ring of polynomials in [Formula: see text] which are integer-valued over the set of matrices [Formula: see text] is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to n.
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spelling pubmed-47509012016-03-02 Integral-valued polynomials over sets of algebraic integers of bounded degree() Peruginelli, Giulio J Number Theory Article Let K be a number field of degree n with ring of integers [Formula: see text]. By means of a criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if [Formula: see text] maps every element of [Formula: see text] of degree n to an algebraic integer, then [Formula: see text] is integral-valued over [Formula: see text] , that is, [Formula: see text]. A similar property holds if we consider the set of all algebraic integers of degree n and a polynomial [Formula: see text]: if [Formula: see text] is integral over [Formula: see text] for every algebraic integer α of degree n, then [Formula: see text] is integral over [Formula: see text] for every algebraic integer β of degree smaller than n. This second result is established by proving that the integral closure of the ring of polynomials in [Formula: see text] which are integer-valued over the set of matrices [Formula: see text] is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to n. Elsevier 2014-04 /pmc/articles/PMC4750901/ /pubmed/26949270 http://dx.doi.org/10.1016/j.jnt.2013.11.007 Text en © 2014 The Author http://creativecommons.org/licenses/by/3.0/ This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
spellingShingle Article
Peruginelli, Giulio
Integral-valued polynomials over sets of algebraic integers of bounded degree()
title Integral-valued polynomials over sets of algebraic integers of bounded degree()
title_full Integral-valued polynomials over sets of algebraic integers of bounded degree()
title_fullStr Integral-valued polynomials over sets of algebraic integers of bounded degree()
title_full_unstemmed Integral-valued polynomials over sets of algebraic integers of bounded degree()
title_short Integral-valued polynomials over sets of algebraic integers of bounded degree()
title_sort integral-valued polynomials over sets of algebraic integers of bounded degree()
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4750901/
https://www.ncbi.nlm.nih.gov/pubmed/26949270
http://dx.doi.org/10.1016/j.jnt.2013.11.007
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