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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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Detalles Bibliográficos
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
Descripción
Sumario: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.