Past exam of the mathematics course of the University of Cambridge / 2007 / ii / Paper 1 / 20H / c / Solution

Modulo five, $X^2+26=(X-2)(X+2)$. Choose ${\mathfrak{p}}_5=(5,2+\alpha )$; then $(5)={\mathfrak{p}}_5{\overline{\mathfrak{p}}}_5$, and $N(2+\alpha )=4+26=30$. The element $2+\alpha$ belongs to ${\mathfrak{p}}_2$, ${\mathfrak{p}}_3$ and ${\mathfrak{p}}_5$. Their product has norm $2\cdot 3\cdot 5=30$, soThe imaginary-quadratic Minkowski bound for ideal classes is $(2/\pi )\sqrt{104}<7$. Every ideal class has an integral representative of norm at most six, whose prime ideal factors lie over two, three or five. Thus these prime ideals generate the ideal class group. Conjugate classes are inverses, and the displayed relation eliminates the class over five. The remaining generators have orders two and three, so their product has order six and generates the whole group: