= Unramified cubic extension of Q of square root minus fifty-nine
{title2=$\mathbb Q(\sqrt{-59})\subseteq\operatorname{Spl}_{\mathbb Q}(T^3+2T+1)$}
The splitting field has <Galois group> $S_3$ and quadratic subfield $\mathbb Q(\sqrt{-59})$. The extension over that subfield is abelian of degree three and unramified everywhere. The only possible rational ramified prime is $59$; at that prime a simple residue root lifts, so the local splitting field is quadratic and coincides with the quadratic subfield's completion. It follows from the <Hilbert class field> degree that $3\mid h_{\mathbb Q(\sqrt{-59})}$.
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