# Rational thinking

Does there exist a polynomial $p\in\mathbb{Q}[X_1,\ldots,X_n]$ such that for any $\alpha_1,\ldots,\alpha_n\in\mathbb{C}$ we have that $p(\alpha_1,\ldots,\alpha_n)\in\mathbb{Q}$ if and only if $\alpha_1,\ldots,\alpha_n\in\mathbb{Q}$?