Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 4 / 20G (Number Fields)

The left side is the norm form $N_{K/\mathbb{Q}}(x+y\alpha +z{\alpha }^2)$ for ${\alpha }^3=7$. Reducing a hypothetical equation modulo 7 and then descending through the forced divisibilities of $x,y,z,t$ gives an infinite descent, so only the zero solution exists. If $N(\theta )=3$, clearing denominators minimally and applying the norm equation would produce a forbidden nonzero integer solution; hence no such $\theta$ exists.

Dedekind's criterion factors $(p)$ according to the factorization of $X^3-7$ modulo $p$ (the monogenic assumption removes the index obstruction). With $\alpha =\sqrt[3]{7}$,Minkowski's bound is below 11, so classes are generated by prime ideals over $2,3,5,7$. Relations from the displayed factorizations and principal ideals reduce them to $\mathfrak{p}=(3,\alpha -1)$ with ${\mathfrak{p}}^3$ principal. Part (a) shows $\mathfrak{p}$ is not principal, so $\mathrm{Cl}(K)\simeq \mathbb{Z}/3\mathbb{Z}$.