Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 2 / 20H / ii

For the ideal class group of the imaginary quadratic field of discriminant minus eighty-four, write $s=\sqrt{-21}$. Here ${\mathcal{O}}_K=\mathbb{Z}[s]$, $D_K=-84$, and the Minkowski bound for ideal classes is $(2/\pi )\sqrt{84}<6$. Thus every class has a representative of norm at most five. The relevant prime ideals areThe first two are ramified: ${\mathfrak{p}}_2^2=(2)$ and ${\mathfrak{p}}_3^2=(3)$. Also ${\mathfrak{p}}_5^2=(2-s)$, as both sides have norm 25 and $s\equiv 2(\mathrm{mod}{\mathfrak{p}}_5)$. The element $3+s$ has norm 30 andThere are no elements of norms 2, 3, 5 or 6, since the element norm is $a^2+21b^2$. Hence the classes of ${\mathfrak{p}}_2$, ${\mathfrak{p}}_3$ and their product are all nontrivial and distinct. The relation makes the last product equal to the class of ${\mathfrak{p}}_5$. Ideals of norm four are principal, and the two primes above five have the same class of order two. The norm bound therefore gives no further classes. Thus