Let $K$ be a local field. There exists a canonical isomorphism (the local Artin map): $\phi_K: K^* \to \text{Gal}(K^{ab}/K)$

where $K^{ab}$ is the maximal abelian extension of $K$. This map satisfies:

1. Existence: Every finite abelian extension $L/K$ corresponds to an open subgroup $N_{L/K}(L^*) \subseteq K^*$
2. Functoriality: Compatible with norms in towers of extensions
3. Explicit: Unramified part corresponds to $\mathcal{O}_K^*$, totally ramified part to powers of uniformizer

The reciprocity law: $L/K$ corresponds bijectively to open finite-index subgroups of $K^*$ via $N_{L/K}(L^*) \leftrightarrow \text{Gal}(L/K)$.

For a local field $K$ with uniformizer $\pi$ and residue field $\mathbb{F}_q$, there exists a formal group $F$ over $\mathcal{O}_K$ such that:

1. Multiplication by $\pi$: $[\pi]_F(x) \equiv x^q \pmod{\pi}$
2. Division points: Adjoining $\pi^n$-torsion points gives totally ramified abelian extensions
3. Explicit reciprocity: The Artin map on totally ramified extensions is given by action on torsion points

This provides explicit generators for all totally ramified abelian extensions, completing the construction of $K^{ab}$.

In a Galois extension $L/K$ of local fields, the higher ramification groups $G_i$ (in upper numbering) satisfy:

1. $G_i$ is trivial for $i > 0$ sufficiently large
2. The jumps in ramification filtration occur at rational values
3. The Hasse-Arf theorem: if $L/K$ is cyclic of degree $p^n$, the jumps are at integers

This refines the structure of the inertia group beyond just its order.

For a local field $K$ with algebraic closure $\bar{K}$: $1 \to I_K \to \text{Gal}(\bar{K}/K) \to \text{Gal}(\bar{\mathbb{F}}_q/\mathbb{F}_q) \to 1$

where $I_K$ is the inertia group. This splits (non-canonically), giving $\text{Gal}(\bar{K}/K) \cong I_K \rtimes \hat{\mathbb{Z}}$.

The inertia group $I_K$ is pro-solvable, with tame quotient $I_K/P_K \cong \prod_{\ell \neq p} \mathbb{Z}_\ell$ where $P_K$ is the wild inertia (pro-$p$-group).