Math 2.0

Field theory and polynomials Definition

Splitting field

Builds on Split polynomialField generated by adjoining elementsField extension

Definition

Let KK be a field and let f∈K[x]f \in K[x] be nonzero. A splitting field of ff over KK is a field extension L/KL/K with these two properties.

  1. The polynomial ff splits over LL, as f=c (x−a1)⋯(x−an)f = c \, (x - a_1) \cdots (x - a_n).
  2. The field LL is generated over KK by the roots: L=K(a1,…,an)L = K(a_1, \dots, a_n).

Remarks

Splitting fields exist and are unique up to isomorphism. The splitting field of x2+1x^2 + 1 over ℝ\mathbb{R} is ℂ\mathbb{C}. The splitting field of x3−2x^3 - 2 over ℚ\mathbb{Q} is ℚ(23,ω)\mathbb{Q}(\sqrt[3]{2}, \omega) with ω=e2πi/3\omega = e^{2\pi i/3}. Its degree over ℚ\mathbb{Q} is 66. The splitting field of xq−xx^q - x over 𝔽p\mathbb{F}_p, for q=pnq = p^n, is the finite field with qq elements.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionField generated by adjoining elements, definition, no evidence levelField generated by ad…definitionSplitting field, definition, no evidence levelSplitting fielddefinition
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
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  6. Unit of a ring definition
  7. Polynomial evaluation definition
  8. Field definition
  9. Root of a polynomial definition
  10. Subfield definition
  11. Split polynomial definition
  12. Field extension definition
  13. Intermediate field definition
  14. Field generated by adjoining elements definition
  15. Splitting field definition

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Further reading

  1. Web https://en.wikipedia.org/wiki/Splitting_field, Definition
About this page
Tag02NZZ1J0FH
URIm2://02NZZ1J0FH
Kinddefinition
WritingW1 stated
MSC12F05
DefinesSplitting field
UsesSplit polynomial, Field generated by adjoining elements, Field extension

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