Problem: Find all polynomials from $\mathbb{R}\to \mathbb{R}$ $f$ with integer coefficients taking irrationals to irrationals.
My attempt: It is clear that the problem statement is equivalent to finding all $p(x)$ over $\mathbb{Z[x]}$ such that $k+p(x)$ does not have any irrational root for any $k \in \mathbb{Q}$. Stated in another way, $p(x)$ does not cut the x-axis at any irrational point when it is vertically translated by any arbrtitrary rational amount. In particular $p(x)$ cuts the x-axis only at rational points, if it cuts at all.
I dont know what to do next. Someone please help.