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.

Solution

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

 Solution (source code)