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

摘要

Let K be a number field of degree n with ring of integers Ok. 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. ∈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) С O_K. A similar property holds if we consider the set of all algebraic integers of degree n and a polynomial f ? Q[X]: if f(a) is integral over Z for every algebraic integer a of degree n, then f(β) is integral over Z 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 Q[X] which are integer-valued over the set of matrices M_n(Z) is equal to the ring of integral- valued polynomials over the set of algebraic integers of degree equal to n.