Math 2.0

definition

Galois 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.

Explanations

L1 human

A Galois extension has as many symmetries over the base field as its degree allows. For these extensions, the subgroups of the Galois group describe all the intermediate fields.

What this node 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 node is. They do not measure its truth.

What this needs

To read this node, read these 27 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

What depends on this

2 nodes use this node, directly or through other nodes.

Sources

  1. Web https://en.wikipedia.org/wiki/Galois_extension, lead section
  2. Web https://en.wikipedia.org/wiki/Galois_extension, Characterization of Galois extensions

What links here

Used by