Is there an example of a number field $K$ for which the genus field of $K$ is contained strictly in the Hilbert class field of $K$?
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7$\begingroup$ Any quadratic field with class group that is not an elementary $2$-group, the canonical example probably being $\mathbb{Q}(\sqrt{-23})$. $\endgroup$– AurelCommented Jan 22, 2018 at 12:43
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$\begingroup$ @Aurel: Why not turn your comment into an answer? $\endgroup$– GH from MOCommented Jan 22, 2018 at 22:23
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$\begingroup$ @GHfromMO I wasnt't sure the question was going to be considered research level. But sure, why not :-) $\endgroup$– AurelCommented Jan 22, 2018 at 22:32
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$\begingroup$ @Aurel: Thanks. Perhaps adding a bit of explanation (class field theory etc.) might help the OP. Strictly speaking, the question is not of research level, but it is certainly of "graduate school level", so I think it is appropriate here. $\endgroup$– GH from MOCommented Jan 22, 2018 at 22:57
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1 Answer
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Let $K$ be a quadratic field with class group $\mathrm{Cl}(K)$ such that $\mathrm{Cl}(K)\neq \mathrm{Cl}(K)[2]$. Since the Galois group of the genus field is isomorphic to $\mathrm{Cl}(K)[2]$ and the one of the Hilbert class field to $\mathrm{Cl}(K)$, this provides an example.
The canonical example is probably $K=\mathbb{Q}(\sqrt{-23})$: it has class number $3$, and its Hilbert class field $H$ is generated over $K$ by a root of $x^3-x-1$. The field $H$ is Galois over $\mathbb{Q}$, but its Galois group is $D_3$, so it is not the compositum of $K$ and a cyclic cubic extension of $\mathbb{Q}$.
