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Canonical static concept record

$p$ -adic field

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Summary

A finite extension of the $p$ -adic numbers $\mathbb Q_p$ .

Record metadata

Canonical name: $p$ -adic field
Concept ID: p_adic_field
Content version: 2.3.0
Primary field: Mathematics
Fields: Mathematics
Primary domain: Number theory
Domains: Number theory, Analysis, Topology
Alternate terminology: No alternate terminology is currently authored for this record.

Carrier(s)

complete valued field $K$

Axioms / constraints

finite extension of $\mathbb Q_p$

Concept sources

Wikipedia — $p$-adic number field — p-adic number field · expository · source ID wp-p-adic-field
nLab — $p$-adic field — p-adic field · reference · source ID nlab-p-adic-field
Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory

Incoming relations (arrows to this concept)

Each relation below ends at this concept.

Completion at a valuation $p$ -adic field

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: story_completion_at_valuation_to_p_adic_field_framework_specialization
Relation type: Framework specialization framework-specialization
Direction: source → target
Endpoint roles: source: Specializes to; target: Special case or refinement of
Authored annotation: non-Archimedean finite extension

Authored explanation

Completing a number field at a non-Archimedean place gives a finite extension of $\mathbb Q_p$ .

How to interpret this relation type

A more specific theory, model, entity class, or regime is obtained by restricting or extending the scope of a broader framework.

Relation sources

Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory

Local field $p$ -adic field

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This is an authored directed relation from the source endpoint to the target endpoint.

Relation ID: e_local_padic
Relation type: Impose axiom impose-axiom
Direction: source → target
Endpoint roles: source: Builds toward; target: Built from
Authored annotation: + characteristic $0$ , residue characteristic $p$

Authored explanation

Restrict to finite extensions of $Q_{p}$ .

How to interpret this relation type

Keep the existing data and select the subclass satisfying an additional law, existence condition, finiteness condition, or other property.

Relation sources

Wikipedia — $p$-adic number field — p-adic number field · expository · source ID wp-p-adic-field
nLab — $p$-adic field — p-adic field · reference · source ID nlab-p-adic-field
Milne — Algebraic Number Theory — Algebraic Number Theory · course text · source ID milne-algebraic-number-theory

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No direct relations are authored in this direction.

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$p$ -adic field

Summary

Record metadata

Carrier(s)

Axioms / constraints

Concept sources

Incoming relations (arrows to this concept)

Completion at a valuation $p$ -adic field

Authored explanation

How to interpret this relation type

Relation sources

Local field $p$ -adic field

Authored explanation

How to interpret this relation type

Relation sources

Outgoing relations (arrows from this concept)