Math 2.0

Field theory and polynomials Definition

Galois extension

Builds on Normal extensionSeparable extension

Definition

An algebraic extension L/KL/K is a Galois extension if it is normal and separable.

Remarks

For a finite extension L/KL/K, these conditions are equivalent.

  1. L/KL/K is Galois.
  2. LL is the splitting field of a separable polynomial over KK.
  3. ∣Aut⁡(L/K)∣=[L:K]|\operatorname{Aut}(L/K)| = [L : K].
  4. KK is the fixed field of Aut⁡(L/K)\operatorname{Aut}(L/K).

Examples: ℂ/ℝ\mathbb{C}/\mathbb{R}, ℚ(2,3)/ℚ\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q} and every extension of finite fields 𝔽pn/𝔽p\mathbb{F}_{p^n}/\mathbb{F}_p. The extension ℚ(23)/ℚ\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q} is not Galois.

Reading path

To understand this page from the ground up, read these 27 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 ringdefinitionSubring, 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 levelFielddefinitionIntegral 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 elementdefinitionMinimal 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 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. Ring definition
  3. Commutative ring definition
  4. Polynomial ring definition
  5. Subring definition
  6. Unit of a ring definition
  7. Zero divisor definition
  8. Degree of a polynomial definition
  9. Formal derivative definition
  10. Polynomial evaluation definition
  11. Field definition
  12. Integral domain definition
  13. Leading coefficient definition
  14. Separable polynomial definition
  15. Root of a polynomial definition
  16. Subfield definition
  17. Irreducible element definition
  18. Monic polynomial definition
  19. Split polynomial definition
  20. Irreducible polynomial definition
  21. Field extension definition
  22. Algebraic element definition
  23. Minimal polynomial of an algebraic element definition
  24. Algebraic extension definition
  25. Separable extension definition
  26. Normal extension definition
  27. Galois extension definition

Where this leads

14 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Galois_extension, lead section
  2. Web https://en.wikipedia.org/wiki/Galois_extension, Characterization of Galois extensions
About this page
Tag0DS2F7GK05
URIm2://0DS2F7GK05
Kinddefinition
WritingW1 stated
MSC12F10
DefinesGalois extension
UsesNormal extension, Separable extension

Markdown · JSON · Edit on GitHub

What this page needs for W2

  • edge instance_of in: one example (0 of 1)
  • edge not_instance_of in: one non-example that fails one clause (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

Used by