Math 2.0

Field theory and polynomials Definition

Abelian extension

Builds on Galois extensionGalois groupAbelian group

Definition

Let L/KL/K be a Galois extension with Galois group Gal⁡(L/K)\Gal(L/K). The extension L/KL/K is abelian if Gal⁡(L/K)\Gal(L/K) is an abelian group.

Remarks

Example: ℚ(2,3)/ℚ\Q(\sqrt{2}, \sqrt{3})/\Q is Galois of degree 44. Its Galois group consists of the four maps 2↦±2\sqrt{2} \mapsto \pm\sqrt{2}, 3↦±3\sqrt{3} \mapsto \pm\sqrt{3}, which commute, so the extension is abelian. Non-example: the splitting field L=ℚ(23,ω)L = \Q(\sqrt[3]{2}, \omega) of x3−2x^3 - 2, with ω=e2πi/3\omega = e^{2\pi i/3}, is Galois over ℚ\Q. Its Galois group permutes the three roots of x3−2x^3 - 2 in all six ways, so it is isomorphic to S3S_3, which is not abelian. Every subextension of a finite abelian extension is abelian, because every subgroup of an abelian group is normal and every quotient is abelian.

Reading path

To understand this page from the ground up, read these 34 pages in order. The foundations come first.

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Binary operation, definition, no evidence levelBinary operationdefinitionGroup, concept, no evidence levelGroupconceptCommutativity, definition, no evidence levelCommutativitydefinitionRing, definition, no evidence levelRingdefinitionAbelian group, definition, no evidence levelAbelian groupdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionPolynomial ring, definition, no evidence levelPolynomial ringdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionZero divisor, definition, no evidence levelZero divisordefinitionDegree of a polynomial, definition, no evidence levelDegree of a polynomialdefinitionFormal derivative, definition, no evidence levelFormal derivativedefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionLeading coefficient, definition, no evidence levelLeading coefficientdefinitionSeparable polynomial, definition, no evidence levelSeparable polynomialdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionIrreducible element, definition, no evidence levelIrreducible elementdefinitionMonic polynomial, definition, no evidence levelMonic polynomialdefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionGalois group, definition, no evidence levelGalois groupdefinitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionSeparable extension, definition, no evidence levelSeparable extensiondefinitionNormal extension, definition, no evidence levelNormal extensiondefinitionGalois extension, definition, no evidence levelGalois extensiondefinitionAbelian extension, definition, no evidence levelAbelian extensiondefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Binary operation definition
  2. Group concept
  3. Commutativity definition
  4. Ring definition
  5. Abelian group definition
  6. Commutative ring definition
  7. Polynomial ring definition
  8. Ring homomorphism definition
  9. Subring definition
  10. Unit of a ring definition
  11. Zero divisor definition
  12. Degree of a polynomial definition
  13. Formal derivative definition
  14. Polynomial evaluation definition
  15. Field definition
  16. Ring isomorphism definition
  17. Integral domain definition
  18. Leading coefficient definition
  19. Separable polynomial definition
  20. Root of a polynomial definition
  21. Subfield definition
  22. Irreducible element definition
  23. Monic polynomial definition
  24. Split polynomial definition
  25. Field extension definition
  26. Irreducible polynomial definition
  27. Algebraic element definition
  28. Galois group definition
  29. Minimal polynomial of an algebraic element definition
  30. Algebraic extension definition
  31. Separable extension definition
  32. Normal extension definition
  33. Galois extension definition
  34. Abelian extension definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Abelian_extension
About this page
TagSR51EVK9H0
URIm2://SR51EVK9H0
Kinddefinition
WritingW0 stub
MSC12F10, 11R20
DefinesAbelian extension
UsesGalois extension, Galois group, Abelian group
Changed2 November 2026

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