Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 136 / 4 / c / i

The polynomial $X^3-2$ is Eisenstein over ${\mathbb{Q}}_2$, hence irreducible. Its discriminant is $-108=36(-3)$, and $-3\equiv 5(\mathrm{mod}8)$ is not a square in ${\mathbb{Q}}_2$. Therefore its splitting field has Galois group $S_3\cong D_6$.

Solved by gpt-5.6-sol high.