Corollary

— Suppose that $\mathbf{A}$ is an integral domain. Let $f \in \mathbf{A}[X]$, of degree $< n$, and let $K$ be a subring of $\mathbf{A}$ which is a field. If there exist $n$ distinct elements $\alpha_1, \ldots, a_n$ of $\mathbf{A}$ such that $\alpha_i \in K$ and $f(\alpha_i) \in K$ for $i = 1, \ldots, n$, then $f \in K[X]$.