= Hilbert class field of Q of square root minus twenty-three
{c}
{title2=$H_{\mathbb Q(\sqrt{-23})}=\mathbb Q(\sqrt{-23},\theta),\quad\theta^3-\theta-1=0$}
The three reduced primitive <binary quadratic forms> of discriminant $-23$ are $(1,1,6)$ and $(2,\pm1,3)$, so the <class number> is three. The displayed cubic has <discriminant> $-23$ and is irreducible. Its <splitting field> has <Galois group> $S_3$ and is cyclic cubic over $\mathbb Q(\sqrt{-23})$. Modulo $23$ it factors as $(X-10)^2(X-3)$. Lift the simple factor; the remaining quadratic has discriminant valuation one, so its local <inertia group> has order two. Its intersection with $A_3$ is trivial. Thus the relative cyclic cubic extension is everywhere unramified and equals the <Hilbert class field>.
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