= Hilbert class field of Q of square root minus thirty-one
{c}
{title2=$H_K=K(\theta),\quad K=\mathbb Q(\sqrt{-31}),\quad\theta^3+\theta-1=0$}
The <splitting field> of $T^3+T-1$ has discriminant square class $-31$ and <Galois group> $S_3$. It is the <Hilbert class field> of $K=\mathbb Q(\sqrt{-31})$: the relative extension is unramified cyclic cubic, and the <ideal class group of Q of square root minus thirty-one> has order three. Locally at $31$, the simple residue root $28$ lifts and the remaining quadratic splits over $\mathbb Q_{31}(\sqrt{-31})$, so the prime of $K$ above $31$ splits completely in its <Hilbert class field>. The inverse generator satisfies $T^3-T^2-1=0$.
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