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Integral-valued polynomials over sets of algebraic integers of bounded degree
1Institut für Analysis und Comput. Number Theory, Technische Universität, Steyrergasse 30, A-8010 Graz, Austria.
Abstract:
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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