= Unramified cyclic cubic extension from a prime-discriminant cubic
{title2=$\operatorname{Gal}(L/\mathbb Q(\sqrt{-p}))\cong C_3$}
Suppose an irreducible monic integral cubic has discriminant $-p$ for a prime $p>3$, and let $L$ be its <splitting field>. Its <Galois group> is $S_3$ and its quadratic subfield is $\mathbb Q(\sqrt{-p})$. The extension over that subfield is cyclic cubic and everywhere unramified. Away from $p$, separable reduction and <Hensel lemma> put the roots in an unramified local extension. At $p$, discriminant <valuation> one gives a double residue root and a separate simple root; lift the simple root, and the remaining quadratic's discriminant has square class $-p$. Its splitting field is already the quadratic subfield's completion. The imaginary quadratic base has no real places.
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