Past exam of the mathematics course of the University of Cambridge / 2021 / ii / Paper 1 / 20G / a / Solution

The element $\alpha$ is a root ofBy the rational root theorem, any rational root of this monic polynomial must be one ofDirect substitution shows that none is a root. A reducible cubic over a field has a linear factor, so $f$ is an irreducible polynomial over $\mathbb{Q}$. It is therefore the minimal polynomial of $\alpha$, and