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Algebraic number field

In mathematics, an algebraic number field (or simply number field) is an extension field K {\displaystyle K} of the field of rational numbers Q {\displaystyle \mathbb {Q} } such that the field extension K / Q {\displaystyle K/\mathbb {Q} } has finite degree (and hence is an algebraic field extension). Thus K {\displaystyle K} is a field that contains Q…

Examples, Algebraicity, and ring of integers & Places

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Overview

Definition

Examples

Algebraicity, and ring of integers

Bases for number fields

Regular representation, trace and discriminant

Places

Ramification

Galois groups and Galois cohomology

Local-global principle

Generalizations

Algebraic number theory

Class field theory

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Algebraic number field

Nodes207
Edges206
Triples31
Avg. degree1.99
Density0.009662
Components1

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Algebraic number field

Top relations

related to Examples · 12
Algebraic number field → Arithmetic, At, Euler, Explicitly, Gaussian, Its, Many, More, Such, The, The Gaussian, This
related to Algebraicity, and ring of integers · 7
Algebraic number field → Every, Generally, Given, In, K/L, Proof, Therefore
related to Prerequisites · 4
Algebraic number field → Another, The, These, To
related to Definition · 2
Algebraic number field → An, Here

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Important terminology

displaystyle field number mathbb algebraic numbers fields place mathcal example element ideal prime ring extension polynomial rational degree integers ultrametric

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
the Frobenius mapinstance ofwhere the mi are all integers.Working locally and using tools0.80text
it is always possible to explicitly compute such a basisinstance ofwhere the mi are all integers.Working locally and using tools0.80text
and it is now standard for computer algebra systems to have built-in programs to do this.Power basisLet Kinstance ofwhere the mi are all integers.Working locally and using tools0.80text
and it is now standard for computer algebra systems to have built-in programs to do thisinstance ofwhere the mi are all integers.Working locally and using tools0.80text
intermediate value theorem at the archimedean placesinstance ofclassical analytic tools0.80text
p-adic analysis at the nonarchimedean placesinstance ofclassical analytic tools0.80text
Algebraic number fieldrelated to Algebraicity, and ring of integersGenerally0.60section
Algebraic number fieldrelated to Algebraicity, and ring of integersK/L0.60section
Algebraic number fieldrelated to Algebraicity, and ring of integersEvery0.60section
Algebraic number fieldrelated to Algebraicity, and ring of integersProof0.60section
Algebraic number fieldrelated to Algebraicity, and ring of integersIn0.60section
Algebraic number fieldrelated to Algebraicity, and ring of integersTherefore0.60section

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