The proof of global Artin reciprocity proceeds through several major steps, requiring deep results from algebraic number theory.

Step 1: Local class field theory

First establish local reciprocity: for local fields $K_v$, construct the local Artin map: $\phi_v: K_v^* \to \text{Gal}(K_v^{ab}/K_v)$

using Lubin-Tate theory (for totally ramified extensions) and explicit construction of unramified extensions (via residue fields). This is more tractable than the global case.

Step 2: Cohomological formulation

Reformulate using group cohomology. The fundamental class in $H^2(\text{Gal}(L/K), L^*)$ provides a canonical element. The Artin map arises from connecting: $H^0(\text{Gal}(L/K), \mathbb{Q}/\mathbb{Z}) \to H^2(\text{Gal}(L/K), L^*)$

This cohomological perspective, developed by Tate and Artin-Tate, simplifies the proof considerably.

Step 3: Product formula

For $\alpha \in K^*$, the product formula states: $\prod_v |\alpha|_v = 1$

where the product is over all places (finite and infinite). This connects local invariants globally.

The sum of local invariants $\sum_v \text{inv}_v(\alpha) = 0$ in $\mathbb{Q}/\mathbb{Z}$ follows, where $\text{inv}_v: H^2(K_v, \bar{K}_v^*) \to \mathbb{Q}/\mathbb{Z}$ is the local invariant map.

Step 4: Global reciprocity from local

Combine local reciprocity laws using the product formula. For an abelian extension $L/K$: $\prod_v \phi_{L_w/K_v}(\alpha) = 1 \in \text{Gal}(L/K)$

for all $\alpha \in K^*$. This product compatibility is the global reciprocity law.

Step 5: Existence theorem

Prove that for every open subgroup $H \subseteq I_K$ of finite index, there exists an abelian extension $L/K$ with $N_{L/K}(I_L) = H$.

Use class formations and limit arguments: construct $L$ as the fixed field of a suitable closed subgroup of $\text{Gal}(K^{ab}/K)$.

Step 6: Uniqueness and isomorphism

Combine existence with reciprocity to establish: $\text{Cl}_\mathfrak{m}(K) / N_{L/K}(\text{Cl}_\mathfrak{m}(L)) \xrightarrow{\Phi_{L/K}} \text{Gal}(L/K)$

is an isomorphism. The kernel identification $\ker(\Phi_{L/K}) = N_{L/K}(I_\mathfrak{m}(L))$ follows from local-global compatibility.

Conclusion: This establishes the complete Artin reciprocity law, classifying all abelian extensions via generalized class groups.