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 OK of degree n to an algebraic integer, then h(X) is integral-valued over OK, that is, h(OK) ⊂ OK. A similar property holds if we consider the set of all algebraic integers of degree n and a polynomial f ∈ ℚ[X]: if f(α) is integral over ℤ for every algebraic integer α 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 Mn(ℤ) is equal to the ring of integral-valued polynomials over the set of algebraic integers of degree equal to n.
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机译：令K为n阶整数环的数字字段。通过吉尔梅准则，一个数域整数环的多项式密集子集，我们证明，如果h∈K[X]将度为OK的OK的每个元素映射到一个代数整数，则h（X）在OK上是整数值，即h（O K ） em> O K 。如果考虑度为 n 的所有代数整数和多项式为 f ∈∈[ℚ X ]的集合，则具有相似的性质：if f （α）对于度为 n 的每个代数整数α，然后为 f n 的度数的每个代数整数β，em>（β）在over上都是积分的。通过证明that [ X ]中多项式环的整数闭环来建立第二个结果，这些多项式环在矩阵 M n上是整数值（ℤ）等于度为 n 的一组代数整数上的整数值多项式的环。
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