Math 2.0

Field theory and polynomials Galois theory Problem

Inverse Galois problem

Builds on Galois extensionGalois group

Problem

Is every finite group GG isomorphic to the Galois group Gal⁡(L/ℚ)\Gal(L/\Q) of some Galois extension LL of the rational numbers ℚ\Q?

Remarks

The problem is open. Every finite abelian group is a Galois group over ℚ\Q, realised inside a cyclotomic extension ℚ(ζn)\Q(\zeta_n). The symmetric groups SnS_n are Galois groups over ℚ\Q; one route uses Hilbert's irreducibility theorem. Every finite solvable group is a Galois group over ℚ\Q, by Shafarevich's theorem. The same question can be asked over other fields; over ℂ(t)\C(t) every finite group is a Galois group. Over a finite field 𝔽q\F_q only cyclic groups occur, since every finite extension of 𝔽q\F_q is cyclic.

Reading path

To understand this page from the ground up, read these 31 pages in order. The foundations come first.

Show the path as a graph
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 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 extensiondefinitionInverse Galois problem, problem, no evidence levelInverse Galois problemproblem
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. Algebraic element definition
  25. Galois group definition
  26. Minimal polynomial of an algebraic element definition
  27. Algebraic extension definition
  28. Separable extension definition
  29. Normal extension definition
  30. Galois extension definition
  31. Inverse Galois problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Inverse_Galois_problem
About this page
TagYJS6X8JN7D
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Kindproblem
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MSC12F12, 11R32
StatusThis copy of the site has no build data. Run m2 status YJS6X8JN7D.
AboutGalois group
UsesGalois extension, Galois group
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Changed2 November 2026

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