Let $L$ be the Hilbert Class Field of $K$, then:
- $Gal(L/K) \cong Cl(K)$ by Artin reciprocity, where $Cl(K)$ is the class group of $K$.
- though being Galois is not transitive in general, we nonetheless have for $K/\mathbb{Q}$ and $L/K$ Galois, that also $L/\mathbb{Q}$ is Galois.
- if $I$ is an ideal in $\mathcal{O}_K$, then $I \mathcal{O}_L$ becomes principal (link)
- a prime $\mathfrak{p}$ of $\mathcal{O}_K$ is principal $\Leftrightarrow \mathfrak{p}$ splits completely in $\mathcal{O}_L$
Since I have only started studying algebraic number theory recently, I still have limited experience. But I was wondering if the HCF has other "well known" nice algebraic properties.