= Ideal class group of Q of square root minus thirty
{c}
{title2=$\operatorname{Cl}(\mathbb Q(\sqrt{-30}))\cong C_2\times C_2$}
Put $\theta=\sqrt{-30}$. The <ring of integers> is $\mathbb Z[\theta]$ and its <discriminant> is $-120$. The <Minkowski bound for ideal classes> $4\sqrt{30}/\pi<7$ shows that the <ideal class group> is generated by the classes of the ramified <prime ideals> $P_2=(2,\theta)$, $P_3=(3,\theta)$ and $P_5=(5,\theta)$. There is no additional <prime ideal> of norm four because two ramifies with residue degree one. The relations $P_i^2=(i)$ and $(\theta)=P_2P_3P_5$ show that these classes have order dividing two and their product is one. None is a <principal ideal>: the element <field norm> $a^2+30b^2$ cannot equal two, three or five. Thus there are exactly four classes, with the displayed group structure. This also supplies the class-number input for the <Hilbert class field of Q of square root minus thirty>.
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