Local field

A local field is a nondiscrete locally compact topological field. A non-Archimedean local field is a finite extension of

Q_{p}

or of

F_{p} ((t))

.

Table of contents

Hensel lemma

For a complete discrete valuation ring

R

, a simple root of a polynomial modulo the maximal ideal lifts uniquely to a root in

R

.

Laurent series field ( $k ((t))$ )

The Laurent series field

k ((t))

consists of formal series

\sum_{n \geq N} a_{n} t^{n}

. Its

t

-adic valuation is the least exponent with nonzero coefficient.

p-adic logarithm ( $lo g_{p}$ )

The

p

-adic logarithm is the convergent series

lo g (1 + x) = \sum_{n \geq 1} (- 1)^{n + 1} x^{n} / n

on a sufficiently small group of principal units.

p-adic exponential ( $exp_{p}$ )

The

p

-adic exponential

exp (x) = \sum_{n \geq 0} x^{n} / n!

converges near zero and is locally inverse to the

p

-adic logarithm.

Teichmuller representative

The Teichmuller representative of a nonzero residue class

\overset{a}{ˉ} \in F_{q}^{*}

is the unique lift

ω

satisfying

ω^{q - 1} = 1

and

ω \equiv a

modulo the maximal ideal.

Unramified extension

A finite extension of local fields is unramified when its ramification index is one and its residue-field degree equals the field degree. A local field has a unique unramified extension of each positive degree inside a fixed algebraic closure.

Ramification group ( $G_{i}$ )

For a finite Galois extension of local fields, the lower ramification groups measure how closely automorphisms fix the valuation ring. The inertia group is

G_{0}

, and

G_{1}

is the wild inertia group.

Inertia group ( $G_{0}$ )

The inertia group of a finite Galois extension of local fields is the kernel of the action of its Galois group on the residue field. In lower numbering it is the zeroth ramification group

G_{0}

.

Wild inertia group ( $G_{1}$ )

The wild inertia group is the first ramification group

G_{1}

. It is the unique Sylow subgroup of the inertia group for the residue characteristic.

Eisenstein polynomial

For a discrete valuation ring with uniformizer

π

, an Eisenstein polynomial is a monic polynomial whose nonleading coefficients are divisible by

π

and whose constant coefficient is not divisible by

π^{2}

. It is irreducible, and adjoining one of its roots gives a totally ramified extension.

Newton polygon

The Newton polygon of a polynomial over a valued field is the lower convex hull of the points formed from coefficient indices and valuations. Its slopes determine the valuations of roots, counted with multiplicity.

Ramification index ( $e (L / K)$ )

For a finite extension

L / K

of non-Archimedean local fields, the ramification index

e (L / K)

is the index of the value group of

K

in that of

L

. Normalized valuations satisfy

v_{L} ∣_{K} = e (L / K) v_{K}

.

Residue-field degree ( $f (L / K)$ )

The residue-field degree of a finite extension of local fields is

f (L / K) = [k_{L} : k_{K}]

, and

[L : K] = e (L / K) f (L / K)

.

Totally ramified extension

A finite extension of local fields is totally ramified when its residue-field degree is one, equivalently when its ramification index equals its field degree.

p-adic unit ( $Z_{p}^{\times}$ )

A p-adic unit is an element of

Z_{p}^{\times}

, equivalently a p-adic integer of valuation zero.

Principal unit

For a non-Archimedean local field

K

with maximal ideal

m_{K}

, a principal unit is an element of

1 + m_{K}

. The higher principal-unit groups are

1 + m_{K}^{n}

.

Roots of unity in a p-adic field ( $μ (K)$ )

The roots of unity in a finite extension

K / Q_{p}

form a finite group. Each element of their prime-to-

p

part is a Teichmuller representative, while any

p

-power part lies among the principal units.

Maximal abelian extension ( $K^{ab}$ )

The maximal abelian extension

K^{ab}

K

is the compositum in a fixed separable closure of all finite abelian extensions of

K

.

Weil group ( $W_{K}$ )

For a non-Archimedean local field

K

, the Weil group is the inverse image of the infinite cyclic subgroup generated by Frobenius under

Gal (\overline{K} / K) \to Gal (\overline{k} / k) ≅ Z

. It is dense in the absolute Galois group with its profinite topology.

Cyclotomic extension of a p-adic field

The cyclotomic extension

Q_{p} (ζ_{p^{n}}) / Q_{p}

is generated by a primitive

p^{n}

th root of unity. It is totally ramified and has Galois group

(Z / p^{n} Z)^{\times}

.

Local Kronecker-Weber theorem

The local Kronecker-Weber theorem states that every finite abelian extension of

Q_{p}

is contained in an extension obtained by adjoining roots of unity.

Local class field theory

Local class field theory describes the abelian extensions of a local field

K

through its multiplicative group

K^{\times}

.

Local Artin reciprocity

Local Artin reciprocity gives a continuous homomorphism

Art_{K} : K^{\times} \to Gal (K^{ab} / K)

with dense image. For every finite abelian extension

L / K

, it induces an isomorphism

K^{\times} / N_{L / K} (L^{\times}) ≅ Gal (L / K) .

Local Artin map ( $Art_{K}$ )

The local Artin map is the reciprocity homomorphism of Local Artin reciprocity. Its normalization is fixed by choosing whether a uniformizer maps to arithmetic or geometric Frobenius.

Norm subgroup of a local field extension ( $N_{L / K} (L^{\times})$ )

For a finite extension of local fields

L / K

, its norm subgroup is the image of the field norm

N_{L / K} : L^{\times} \to K^{\times}

. For an abelian extension, it is the kernel of the induced local Artin map to

Gal (L / K)

.

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The term "local field" can refer to different concepts in different contexts, including mathematics, physics, and other fields. Here are two common meanings: 1. **Local Fields in Number Theory**: In the context of algebraic number theory, a local field is a complete field with respect to a discrete valuation, which is often associated with the study of numbers in number fields. These fields are typically used to examine the local properties of arithmetic objects.

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