Math 2.0

Field theory and polynomials Definition

Cyclic extension

Builds on Galois extensionGalois groupCyclic 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 cyclic if Gal⁡(L/K)\Gal(L/K) is a cyclic group. It is cyclic of degree nn if moreover Gal⁡(L/K)\Gal(L/K) has order nn.

Remarks

Example: let pp be a prime, qq a power of pp, and n≥1n \ge 1. The extension 𝔽qn/𝔽q\F_{q^n}/\F_q of finite fields is cyclic of degree nn, generated by the Frobenius map x↦xqx \mapsto x^q. Non-example: ℚ(2,3)/ℚ\Q(\sqrt{2}, \sqrt{3})/\Q is Galois with group ℤ/2×ℤ/2\Z/2 \times \Z/2. Every element of this group has order at most 22, while the group has order 44, so it is not cyclic. Every cyclic extension is an abelian extension.

Reading path

To understand this page from the ground up, read these 35 pages in order. The foundations come first.

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Group, concept, no evidence levelGroupconceptLaws of exponents in a group, statement, no evidence levelLaws of exponents in …statementRing, definition, no evidence levelRingdefinitionSubgroup, definition, no evidence levelSubgroupdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionCyclic subgroup generated by an element, definition, no evidence levelCyclic subgroup gener…definitionPolynomial 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 divisordefinitionCyclic group, definition, no evidence levelCyclic groupdefinitionDegree 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 levelFielddefinitionIntegral domain, definition, no evidence levelIntegral domaindefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionLeading 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 polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionAlgebraic 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 extensiondefinitionCyclic extension, definition, no evidence levelCyclic 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. Group concept
  2. Laws of exponents in a group statement
  3. Ring definition
  4. Subgroup definition
  5. Commutative ring definition
  6. Cyclic subgroup generated by an element definition
  7. Polynomial ring definition
  8. Ring homomorphism definition
  9. Subring definition
  10. Unit of a ring definition
  11. Zero divisor definition
  12. Cyclic group definition
  13. Degree of a polynomial definition
  14. Formal derivative definition
  15. Polynomial evaluation definition
  16. Field definition
  17. Integral domain definition
  18. Ring isomorphism definition
  19. Leading coefficient definition
  20. Separable polynomial definition
  21. Root of a polynomial definition
  22. Subfield definition
  23. Irreducible element definition
  24. Monic polynomial definition
  25. Split polynomial definition
  26. Irreducible polynomial definition
  27. Field extension definition
  28. Algebraic element definition
  29. Galois group definition
  30. Minimal polynomial of an algebraic element definition
  31. Algebraic extension definition
  32. Separable extension definition
  33. Normal extension definition
  34. Galois extension definition
  35. Cyclic extension definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Abelian_extension
About this page
TagPWQ031W07T
URIm2://PWQ031W07T
Kinddefinition
WritingW0 stub
MSC12F10
DefinesCyclic extension
UsesGalois extension, Galois group, Cyclic group
Changed2 November 2026

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