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 $K$?



The Genus field and Hilbert class field
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 $\mathbb{Q}$.


