Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 4 / 2G / a / Solution

For each degree and coefficient bound there are finitely many integer polynomials. Hence $\mathbb{Z}[x]$ is countable. Every nonzero polynomial has finitely many roots, and every algebraic number is a root of one of these polynomials. A countable union of finite sets is countable.