Math 2.0

Field theory and polynomials Galois theory Theorem

Fundamental theorem of Galois theory

Builds on Galois extensionGalois groupFixed fieldIntermediate fieldOrder of the Galois group of a Galois extension equals its degree

Statement

Let L/KL/K be a finite Galois extension with Galois group G=Gal⁡(L/K)G = \operatorname{Gal}(L/K).

  1. The maps H↦LHH \mapsto L^H and E↦Gal⁡(L/E)E \mapsto \operatorname{Gal}(L/E) are inverse bijections between the subgroups HH of GG and the intermediate fields EE of L/KL/K.
  2. These bijections reverse inclusions: H1⊆H2H_1 \subseteq H_2 if and only if LH1⊇LH2L^{H_1} \supseteq L^{H_2}.
  3. For every subgroup HH, [L:LH]=∣H∣[L : L^H] = |H| and [LH:K]=∣G∣/∣H∣[L^H : K] = |G|/|H|.
  4. The extension LH/KL^H/K is normal if and only if HH is a normal subgroup of GG. Then Gal⁡(LH/K)≅G/H\operatorname{Gal}(L^H/K) \cong G/H.

Here LHL^H is the fixed field of HH.

Remarks

For L=ℚ(2,3)L = \mathbb{Q}(\sqrt{2}, \sqrt{3}) and K=ℚK = \mathbb{Q}, the group GG has order 44 and three subgroups of order 22. They match the three intermediate fields ℚ(2)\mathbb{Q}(\sqrt{2}), ℚ(3)\mathbb{Q}(\sqrt{3}) and ℚ(6)\mathbb{Q}(\sqrt{6}). The top field LL matches the trivial subgroup, and the base field KK matches GG. A finite Galois extension has only finitely many intermediate fields, because a finite group has finitely many subgroups. The theorem is used to prove that the general equation of degree 55 is not solvable by radicals.

History

Évariste Galois proved the theorem in his development of Galois theory.

Reading path

To understand this page from the ground up, read these 37 pages in order. The foundations come first.

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Group, 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 groupdefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionMinimal polynomial of an algebraic element, definition, no evidence levelMinimal polynomial of…definitionAlgebraic extension, definition, no evidence levelAlgebraic extensiondefinitionFinite extension, definition, no evidence levelFinite extensiondefinitionSeparable extension, definition, no evidence levelSeparable extensiondefinitionNormal extension, definition, no evidence levelNormal extensiondefinitionThe Galois group has order at most the degree, statement, no evidence levelThe Galois group has …statementGalois extension, definition, no evidence levelGalois extensiondefinitionOrder of the Galois group of a Galois extension equals its degree, statement, no evidence levelOrder of the Galois g…statementFundamental theorem of Galois theory, statement, no evidence levelFundamental theorem o…statement
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. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Ring homomorphism definition
  6. Subring definition
  7. Unit of a ring definition
  8. Zero divisor definition
  9. Degree of a polynomial definition
  10. Formal derivative definition
  11. Polynomial evaluation definition
  12. Field definition
  13. Ring isomorphism definition
  14. Integral domain definition
  15. Leading coefficient definition
  16. Separable polynomial definition
  17. Root of a polynomial definition
  18. Subfield definition
  19. Irreducible element definition
  20. Monic polynomial definition
  21. Split polynomial definition
  22. Irreducible polynomial definition
  23. Field extension definition
  24. Fixed field definition
  25. Algebraic element definition
  26. Galois group definition
  27. Degree of a field extension definition
  28. Intermediate field definition
  29. Minimal polynomial of an algebraic element definition
  30. Algebraic extension definition
  31. Finite extension definition
  32. Separable extension definition
  33. Normal extension definition
  34. The Galois group has order at most the degree statement
  35. Galois extension definition
  36. Order of the Galois group of a Galois extension equals its degree statement
  37. Fundamental theorem of Galois theory statement

Where this leads

1 page builds on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory, Explicit description of the correspondence
  2. Web https://en.wikipedia.org/wiki/Fundamental_theorem_of_Galois_theory, Properties of the correspondence
About this page
Tag0T1Q8D0FW8
URIm2://0T1Q8D0FW8
Kindstatement, theorem
WritingW1 stated
MSC12F10
StatusThis copy of the site has no build data. Run m2 status 0T1Q8D0FW8.
UsesGalois extension, Galois group, Fixed field, Intermediate field, Order of the Galois group of a Galois extension equals its degree
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