= Ideal class group of the imaginary quadratic field of discriminant minus eighty-four
{title2=$\operatorname{Cl}(\mathbb Q(\sqrt{-21}))\cong C_2\times C_2$}
For $s=\sqrt{-21}$, the integer ring is $\mathbb Z[s]$. The <Minkowski bound for ideal classes> is less than six. The ramified primes $(2,1+s)$ and $(3,s)$ give independent classes of order two; their product is the class of $(5,s-2)$ because $(3+s)$ factors as the product of these three prime ideals. Element norms $a^2+21b^2$ cannot equal 2, 3 or 6, so these classes are distinct. Ideals of norm at most five supply no further classes, proving the four-element group.
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