polynomials and symmetric functions Suppose I have a polynomial function $f\in \mathbb{Z}[x_1, \dotsc, x_k],$ such that whenever $r_1, \dotsc, r_k$ are roots of a monic polynomial of degree $k$ with integer coefficients, we have $f(r_1, \dotsc, r_k) \in \mathbb{Z}.$ Is it true that $f$ is a symmetric function of its arguments?

$f$ is symmetric in its arguments whenever the Galois group of $r_1, \dots r_k$ is the full symmetric group $S_k$. The set of tuples of algebraic integers with this property is almost certainly Zariski dense in $\mathbb{A}^n$ (although I can't prove it off the top of my head), and so if two polynomials (namely $f$ and one of its permutations) are equal on this set of tuples then they're identical. 
