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Field theory and polynomials Definition

Galois theory

Builds on Field extensionGalois extensionGalois groupFixed fieldIntermediate fieldGalois connection between subgroups and subfields

Definition

Let L/KL/K be a field extension, and let Aut⁡(L/K)\Aut(L/K) be the group of field automorphisms of LL that fix every element of KK. Galois theory is the study of field extensions L/KL/K through the groups Aut⁡(L/K)\Aut(L/K) and the Galois connection between subgroups and subfields. Its basic objects are:

  1. the Galois extensions, that is, the algebraic extensions that are normal and separable;
  2. their Galois groups Gal⁡(L/K)=Aut⁡(L/K)\Gal(L/K) = \Aut(L/K);
  3. the maps H↦LHH \mapsto L^H, from subgroups of Gal⁡(L/K)\Gal(L/K) to their fixed fields, and E↦Gal⁡(L/E)E \mapsto \Gal(L/E), from intermediate fields to subgroups.

Remarks

Example: for L=ℚ(i)L = \Q(i) over ℚ\Q, the group Gal⁡(L/ℚ)\Gal(L/\Q) has two elements, the identity and complex conjugation, and the fixed field of the whole group is ℚ\Q. Non-example of a Galois extension: ℚ(23)/ℚ\Q(\sqrt[3]{2})/\Q is separable but not normal, since x3−2x^3 - 2 has only one root in it. Its automorphism group is trivial, so the base field ℚ\Q is not the fixed field of any subgroup. The Galois group of a separable polynomial f∈K[x]f \in K[x] is the Galois group of a splitting field of ff over KK.

Reading path

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

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Galois connection between subgroups and subfields, concept, no evidence levelGalois connection bet…conceptGroup, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative 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 polynomialdefinitionIrreducible polynomial, definition, no evidence levelIrreducible polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionFixed field, definition, no evidence levelFixed fielddefinitionAlgebraic element, definition, no evidence levelAlgebraic elementdefinitionGalois group, definition, no evidence levelGalois groupdefinitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionMinimal 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 extensiondefinitionGalois theory, definition, no evidence levelGalois theorydefinition
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. Galois connection between subgroups and subfields concept
  2. Group concept
  3. Ring definition
  4. Commutative ring definition
  5. Polynomial ring definition
  6. Ring homomorphism definition
  7. Subring definition
  8. Unit of a ring definition
  9. Zero divisor definition
  10. Degree of a polynomial definition
  11. Formal derivative definition
  12. Polynomial evaluation definition
  13. Field definition
  14. Ring isomorphism definition
  15. Integral domain definition
  16. Leading coefficient definition
  17. Separable polynomial definition
  18. Root of a polynomial definition
  19. Subfield definition
  20. Irreducible element definition
  21. Monic polynomial definition
  22. Split polynomial definition
  23. Irreducible polynomial definition
  24. Field extension definition
  25. Fixed field definition
  26. Algebraic element definition
  27. Galois group definition
  28. Intermediate field 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. Galois theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Galois_theory
About this page
TagBMGE1ZAFQM
URIm2://BMGE1ZAFQM
Kinddefinition
WritingW0 stub
MSC12F10
DefinesGalois theory
UsesField extension, Galois extension, Galois group, Fixed field, Intermediate field, Galois connection between subgroups and subfields
Changed2 November 2026

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