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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.

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In addition to the above properties, The Hilbert class field $H=H_{1}$ of a number field $K$ is the maximal abelian unramified extension of $K$. The generalization of Hilbert Class field is the ray class field $H_{\mathfrak{m}}$ for given any modulus $\mathfrak{m}$ and when $\mathfrak{m}=1$ the corresponding ray class field becomes the Hilbert Class field.

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The root discriminant of a number field $K$ is $$\operatorname{rd}(K) = |d(K)|^{1/n},$$ where $d(K)$ is the discriminant of the field, and $n$ is the degree of $K$ over $\mathbb{Q}$.

One useful property of the Hilbert class field $H(K)$ of $K$ is that it has the same root discriminant of $K$, i.e. $$\operatorname{rd}(K) = \operatorname{rd}(H(K)).$$

Actually this property is true of any extension $L$ of $K$ that is unramified at all the finite primes.

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