= Ideal class group of Q of square root of minus twenty-two
{c}
{title2=$\operatorname{Cl}(\mathbb Q(\sqrt{-22}))\cong C_2$}
The ring of integers is $\mathbb Z[\sqrt{-22}]$, its discriminant is $-88$, and its <Minkowski bound for ideal classes> is $4\sqrt{22}/\pi<6$. The primes three and five are inert, while
$$
(2)=\mathfrak p_2^2,
\qquad
\mathfrak p_2=(2,\sqrt{-22}).
$$
The ideal $\mathfrak p_2$ is nonprincipal because $a^2+22b^2=2$ has no integer solution. Every class is represented by an ideal of norm at most five, so $[\mathfrak p_2]$ is the only possible nontrivial class and the class group is $C_2$.
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