Let $L/K$ be a finite abelian extension with conductor $\mathfrak{m}$. The Artin map: $\Phi_{L/K}: I_\mathfrak{m}(K) \to \text{Gal}(L/K)$

satisfies:

1. Surjectivity: $\Phi_{L/K}$ is surjective
2. Kernel: $\ker(\Phi_{L/K}) = N_{L/K}(I_\mathfrak{m}(L))$
3. Reciprocity Isomorphism: Induces isomorphism $\text{Cl}_\mathfrak{m}(K) / N_{L/K}(\text{Cl}_\mathfrak{m}(L)) \xrightarrow{\sim} \text{Gal}(L/K)$

This completely determines the Galois group in terms of ideal class groups and norms.

Existence: For every open subgroup $H \subseteq I_K$ of finite index containing $K^{*,+}$ (totally positive units), there exists a unique abelian extension $L/K$ with $N_{L/K}(I_L) = H$.

Uniqueness: Two abelian extensions $L_1, L_2$ of $K$ coincide if and only if $N_{L_1/K}(I_{L_1}) = N_{L_2/K}(I_{L_2})$.

This establishes a bijection: $\{\text{abelian extensions of } K\} \leftrightarrow \{\text{open finite-index subgroups of } I_K\}$

In the Hilbert class field $H_K$, every ideal of $K$ becomes principal. Equivalently: $N_{H_K/K}(I_{H_K}) = P_K^+$

where $P_K^+$ denotes principal ideals generated by totally positive elements.

Moreover, $\text{Cl}(H_K) = 1$, so $\mathcal{O}_{H_K}$ is a principal ideal domain.

The conductor $\mathfrak{f}_{L/K}$ of an abelian extension $L/K$ is the product: $\mathfrak{f}_{L/K} = \prod_\mathfrak{p} \mathfrak{p}^{f_\mathfrak{p}}$

where $f_\mathfrak{p}$ is determined by ramification data. The discriminant satisfies: $\mathfrak{d}_{L/K} = \prod_\mathfrak{p} \mathfrak{p}^{d_\mathfrak{p}}$

with $d_\mathfrak{p} = e_\mathfrak{p} - 1 + \delta_\mathfrak{p}$ where $\delta_\mathfrak{p}$ measures wild ramification.

The conductor divides the discriminant: $\mathfrak{f}_{L/K} | \mathfrak{d}_{L/K}$.

An element $\alpha \in K^*$ is a norm from an abelian extension $L/K$ if and only if $\alpha$ is a local norm at every place: $\alpha \in N_{L/K}(L^*) \Leftrightarrow \alpha \in N_{L_v/K_v}(L_v^*) \text{ for all } v$

This local-global principle for norms is a powerful consequence of class field theory, enabling reduction of global questions to local computations.