= Ideal class group of Q of square root minus thirty-four
{c}
{title2=$\operatorname{Cl}(\mathbb Q(\sqrt{-34}))\cong C_4$}
Write $\eta=\sqrt{-34}$. The <ring of integers> is $\mathbb Z[\eta]$ with discriminant $-136$. The <Minkowski bound for ideal classes> is less than eight, so classes of prime ideals of norm at most seven generate the <ideal class group>. These are the ramified prime $P_2=(2,\eta)$ and conjugate pairs at five and seven; three is inert with prime norm nine. Set $P_5=(5,\eta-1)$ and $P_7=(7,\eta+1)$. Conjugate prime classes are inverse, so these three classes suffice.
The <field norm> identities $N(4+\eta)=50$ and $N(1+\eta)=35$ give $(4+\eta)=P_2P_5^2$ and $(1+\eta)=\bar P_5P_7$, by reduction modulo the indicated primes and equality of norms. Thus $[P_5]^2=[P_2]$ and $[P_7]=[P_5]$. Since $P_2^2=(2)$ but $a^2+34b^2=2$ has no integer solution, $[P_2]$ has order two. Therefore $[P_5]$ has order four and generates the entire group, proving the displayed isomorphism.
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