Unramified cubic extension of Q of square root minus fifty-nine ($\mathbb{Q}(\sqrt{-59})\subseteq {\mathrm{Spl}}_{\mathbb{Q}}(T^3+2T+1)$)

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 $3\mid h_{\mathbb{Q}(\sqrt{-59})}$.