= Ray class field of Q of square root minus three modulo seven
{title2=$K_{(7)}=\mathbb Q(\zeta_{21}),\quad K=\mathbb Q(\sqrt{-3})$}
The <ring of integers> is $\mathbb Z[\omega]$, where $\omega^2+\omega+1=0$, and its <class number> is one by the <Minkowski bound for ideal classes>. Modulo seven it is $\mathbb F_7\times\mathbb F_7$. The six <roots of unity> reduce to the kernel of the product map on the two residue <unit groups>, so the <ray class group> is $\mathbb F_7^\times$, of order six. The relative <Artin symbol> on $K(\zeta_7)$ is exponentiation by the <norm of a fractional ideal> modulo seven. It kills the principal ray subgroup, and $[K(\zeta_7):K]=6$, proving that this is the <ray class field>.
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