Local Fields and Valuations - Core Definitions

Local fields are completions of number fields at primes, providing the foundation for local-global principles and $p$ -adic analysis.

DefinitionValuation

A valuation on a field $K$ is a function $v : K^* \to \mathbb{R}$ satisfying:

$v(xy) = v(x) + v(y)$ (multiplicativity)
$v(x + y) \geq \min(v(x), v(y))$ (ultrametric inequality)
$v$ extends to $K$ by $v(0) = \infty$

Two valuations are equivalent if they induce the same topology on $K$ . The valuation ring is $\mathcal{O}_v = \{x \in K : v(x) \geq 0\}$ .

ExampleClassical Valuations

$p$ -adic valuation on $\mathbb{Q}$ : $v_p(p^a m/n) = a$ where $\gcd(p, mn) = 1$
Archimedean valuation: $v_\infty(x) = -\log|x|$ (extends to complex absolute value)
Discrete valuations: $v(\mathfrak{p}) = 1$ for prime $\mathfrak{p}$ in a Dedekind domain

These exemplify the dichotomy: nonarchimedean (ultrametric) vs. archimedean valuations.

DefinitionLocal Field

A local field is a field complete with respect to a discrete valuation and having finite residue field.

Equivalently, $K$ is:

The fraction field of a complete discrete valuation ring (DVR)
With finite residue field $\kappa = \mathcal{O}_K/\mathfrak{m}$ where $\mathfrak{m}$ is the maximal ideal

Every local field is either:

Characteristic 0: finite extension of $\mathbb{Q}_p$ (the $p$ -adic numbers)
Positive characteristic: isomorphic to $\mathbb{F}_q((t))$ (Laurent series over finite field)

ExampleLocal Fields

$\mathbb{Q}_p$ : completion of $\mathbb{Q}$ at $(p)$ , residue field $\mathbb{F}_p$
$\mathbb{Q}_p(\sqrt[3]{p})$ : unramified extension of degree 3
$\mathbb{Q}_p(\zeta_p)$ : totally ramified extension
$\mathbb{F}_p((t))$ : Laurent series over $\mathbb{F}_p$ , characteristic $p$
Completions $K_\mathfrak{p}$ of number fields at primes $\mathfrak{p}$

DefinitionRamification in Local Fields

For a finite extension $L/K$ of local fields with ramification index $e$ and inertia degree $f$ : $[L : K] = ef$

The extension is:

Unramified if $e = 1$ (residue field extension is separable)
Totally ramified if $f = 1$ (no change in residue field)
Tamely ramified if $e$ is coprime to the residue characteristic

Every extension decomposes uniquely as unramified followed by totally ramified.

Remark

Local fields provide a "microscope" for studying number fields at individual primes. Local class field theory (simpler than global) was understood first and guided development of global theory.

Hensel's lemma, a key tool in $p$ -adic analysis, allows lifting solutions modulo powers of $p$ to genuine solutions, enabling explicit computations.