Ideal class group of the imaginary quadratic field of discriminant minus eighty-four ($\mathrm{Cl}(\mathbb{Q}(\sqrt{-21}))\cong C_2\times C_2$)

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.

Parity and valuations at 3 and 7 make the ideals $(y+\sqrt{-21})$ and $(y-\sqrt{-21})$ coprime in any hypothetical solution. Their product is a cube, so the first is an ideal cube. The ideal class group has exponent two, forcing its cube root ideal to be principal. Cubing $a+b\sqrt{-21}$ would then give coefficient $3b(a^2-7b^2)$ on $\sqrt{-21}$, which cannot equal one. This proves the nonexistence of integer solutions without assuming unique element factorization.