For $K = \mathbb{Q}(\sqrt{d})$, Gauss's genus theory describes part of the class group. The genus field $K^{(2)}$ is the maximal unramified abelian extension of exponent dividing 2.

For $d = -105 = -3 \cdot 5 \cdot 7$: The genus group has order $2^2 = 4$, giving intermediate fields: $\mathbb{Q}(\sqrt{-105}) \subset \mathbb{Q}(\sqrt{-3}, \sqrt{5}) = K^{(2)} \subset H_K$

The full Hilbert class field $H_K$ has degree 8 over $K$, containing $K^{(2)}$ of degree 4.

For imaginary quadratic $K = \mathbb{Q}(\sqrt{d})$ with $d < 0$, the Hilbert class field is generated by $j$-invariants of elliptic curves with CM by $\mathcal{O}_K$.

For $K = \mathbb{Q}(i)$: $j(i) = 1728$ generates $H_K = K$ (since $h_K = 1$). For $K = \mathbb{Q}(\sqrt{-14})$: $j(\tau)$ for suitable $\tau$ generates $H_K$, a degree 4 extension.

This complex multiplication theory provides explicit generators for Hilbert class fields of imaginary quadratic fields.

The $n$-th cyclotomic field $\mathbb{Q}(\zeta_n)$ contains many subfields corresponding to subgroups of $(\mathbb{Z}/n\mathbb{Z})^*$.

For $n = 12$: $\text{Gal}(\mathbb{Q}(\zeta_{12})/\mathbb{Q}) \cong (\mathbb{Z}/12\mathbb{Z})^* \cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.

Intermediate fields include $\mathbb{Q}(i), \mathbb{Q}(\sqrt{3}), \mathbb{Q}(\sqrt{-3})$, all abelian over $\mathbb{Q}$ via Kronecker-Weber.

The Artin map for $\mathbb{Q}(\zeta_p)/\mathbb{Q}$ satisfies: $\phi_{\mathbb{Q}(\zeta_p)/\mathbb{Q}}(q) = \text{Frob}_q \in \text{Gal}(\mathbb{Q}(\zeta_p)/\mathbb{Q}) \cong (\mathbb{Z}/p\mathbb{Z})^*$

where $\text{Frob}_q(\zeta_p) = \zeta_p^q$. This gives an explicit isomorphism from prime ideals to Galois elements.

For quadratic extensions $\mathbb{Q}(\sqrt{d})/\mathbb{Q}$: The Artin symbol is the Legendre symbol: $\phi_{\mathbb{Q}(\sqrt{d})/\mathbb{Q}}((p)) = \begin{cases} \text{id} & \text{if } \left(\frac{d}{p}\right) = 1 \\ \sigma & \text{if } \left(\frac{d}{p}\right) = -1 \end{cases}$

where $\sigma$ is the nontrivial automorphism.

For $K = \mathbb{Q}$ and modulus $\mathfrak{m} = (m)$, the ray class field is: $K_\mathfrak{m} = \mathbb{Q}(\zeta_m)$

with $\text{Gal}(K_\mathfrak{m}/K) \cong (\mathbb{Z}/m\mathbb{Z})^*$. Every abelian extension of $\mathbb{Q}$ appears as a subfield of some $\mathbb{Q}(\zeta_m)$ (Kronecker-Weber).

For general $K$, ray class fields are more complicated, but cyclotomic extensions still play a fundamental role via Kummer theory.