= Hilbert class field of Q of square root 229
{c}
{title2=$H=\mathbb Q(\sqrt{229},\alpha),\qquad\alpha^3-4\alpha-1=0$}
The polynomial is irreducible with <discriminant> $229$, so its splitting field has <Galois group> $S_3$ and quadratic subfield $\mathbb Q(\sqrt{229})$. The splitting field is cyclic cubic over that subfield. It is unramified outside 229, and at 229 its inertia is a transposition, meeting the alternating cubic subgroup trivially. All roots are real. It is therefore everywhere <unramified> over the quadratic field. The <ideal class group of Q of square root 229> has order three, so this extension is exactly the <Hilbert class field>.
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