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

Let $K$ be a number field of degree $n$ with ring of integers $O_K$. By means of criterion of Gilmer for polynomially dense subsets of the ring of integers of a number field, we show that, if $hin K[X]$ maps every element of $O_K$ of degree $n$ to an algebraic integer, then $h(X)$ is integral-valued over $O_K$, that is $h(O_K)subset O_K$. A similar property holds if we consider the set of all algebraic integers of degree $n$ and a polynomial $finQ[X]$: if $f(alpha)$ is integral over $Z$ for every algebraic integer $alpha$ of degree $n$, then $f(eta)$ is integral over $Z$ for every algebraic integer $eta$ 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 $n$.