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Field theory and polynomials Definition

Infinite Galois theory

Builds on Galois extensionGalois groupKrull topologyIntermediate fieldFixed fieldAlgebraic extension

Definition

Let L/KL/K be a Galois extension, that is, an algebraic extension that is normal and separable, of possibly infinite degree. Infinite Galois theory is the study of L/KL/K through its Galois group G=Gal⁡(L/K)G = \Gal(L/K) with the Krull topology. Its central statement is the Galois correspondence:

  1. the maps H↦LHH \mapsto L^H, with LHL^H the fixed field of HH, and E↦Gal⁡(L/E)E \mapsto \Gal(L/E) are inverse bijections that reverse inclusion. They match the closed subgroups HH of GG with the intermediate fields EE of L/KL/K;
  2. a closed subgroup HH is open if and only if LH/KL^H/K is finite;
  3. a closed subgroup HH is normal if and only if LH/KL^H/K is Galois, and then Gal⁡(LH/K)≅G/H\Gal(L^H/K) \cong G/H as topological groups.

Remarks

Example: let K=𝔽pK = \F_p and LL an algebraic closure of 𝔽p\F_p. The intermediate fields are the fields 𝔽pn\F_{p^n} and the infinite unions of such fields, and 𝔽pn\F_{p^n} corresponds to the open subgroup generated topologically by ϕn\phi^n, where ϕ(x)=xp\phi(x) = x^p. Non-example of the correspondence for all subgroups: the subgroup H={ϕk:k∈ℤ}H = \set{\phi^k : k \in \Z} is not closed in Gal⁡(L/𝔽p)\Gal(L/\F_p). Its fixed field is 𝔽p\F_p, the same as for the whole group, so H↦LHH \mapsto L^H is not injective on all subgroups. When L/KL/K is finite, the Krull topology is discrete, every subgroup is closed, and the theory reduces to the fundamental theorem of Galois theory.

Reading path

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

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Group, concept, no evidence levelGroupconceptKrull topology, concept, no evidence levelKrull topologyconceptRing, 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 extensiondefinitionInfinite Galois theory, definition, no evidence levelInfinite Galois 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. Group concept
  2. Krull topology 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. Infinite Galois theory definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Galois_extension
About this page
TagNPY85FYJHF
URIm2://NPY85FYJHF
Kinddefinition
WritingW0 stub
MSC12F10, 20E18
DefinesInfinite Galois theory
UsesGalois extension, Galois group, Krull topology, Intermediate field, Fixed field, Algebraic extension
Changed2 November 2026

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