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Splitting field

In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial splits, i.e., decomposes into linear factors. For example, the polynomial x 2 − 2 {\displaystyle x^{2}-2} does not factor over the rationals; its splitting field is the smallest extension of the…

Examples, Constructing splitting fields & Properties

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Overview

Definition

Properties

Constructing splitting fields

Examples

Advanced semantic analysis

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Map overview Semantic statistics

Splitting field

Nodes61
Edges60
Triples47
Avg. degree1.97
Density0.032787
Components1

How this topic connects Entity context

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Splitting field

Top relations

related to References · 17
Splitting field → Abstract Algebra, David, Dummit, EMS Press, Encyclopedia, Eric, Foote, Inc, ISBN, John Wiley, Mathematics, MathWorld, New York, Richard, Sons, Splitting, Weisstein
related to The construction · 8
Splitting field → Choose, Construct, Factorize, Ki, Let, Repeat, Since, The
related to Other examples · 7
Splitting field → F49, F7, Fp, Fq, GF, Sometimes, The
related to Cubic example · 6
Splitting field → Each, It, Let, Such, Therefore, Thus
related to Definition · 4
Splitting field → Galois, It, The, X-
related to Properties · 4
Splitting field → An, Given, If, On
is a · 1
Splitting field → smallest extension of the rationals containing 2

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Important terminology

field splitting displaystyle polynomial x2 extension ki irreducible given ring roots elements quotient root factors containing example degree x3 complex

Entity relationships Subject–Predicate–Object triples

SubjectPredicateObjectConfidenceSrc
Splitting fieldis asmallest extension of the rationals containing 20.90text
Splitting fieldrelated to Cubic exampleLet0.60section
Splitting fieldrelated to Cubic exampleEach0.60section
Splitting fieldrelated to Cubic exampleTherefore0.60section
Splitting fieldrelated to Cubic exampleSuch0.60section
Splitting fieldrelated to Cubic exampleIt0.60section
Splitting fieldrelated to Cubic exampleThus0.60section
Splitting fieldrelated to DefinitionX-0.60section
Splitting fieldrelated to DefinitionThe0.60section
Splitting fieldrelated to DefinitionIt0.60section
Splitting fieldrelated to DefinitionGalois0.60section
Splitting fieldrelated to Other examplesThe0.60section

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    Min side: 3
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