Math 2.0

statement theorem

Order of the Galois group of a Galois extension equals its degree

Statement

Let L/KL/K be a finite Galois extension. Then ∣Gal⁡(L/K)∣=[L:K].|\operatorname{Gal}(L/K)| = [L : K].

Remarks

Conversely, a finite extension with ∣Aut⁡(L/K)∣=[L:K]|\operatorname{Aut}(L/K)| = [L : K] is Galois. So the general bound is an equality exactly for Galois extensions. For Q(2,3)/Q\mathbb{Q}(\sqrt{2}, \sqrt{3})/\mathbb{Q}, the Galois group has order 44: an automorphism can change the signs of 2\sqrt{2} and of 3\sqrt{3} independently. For Fpn/Fp\mathbb{F}_{p^n}/\mathbb{F}_p, the Galois group is cyclic of order nn, generated by the Frobenius automorphism.

Explanations

L1 human

For a Galois extension, counting symmetries is the same as computing the degree. This links the size of a group to the size of a field extension.

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To read this node, read these 34 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 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 groupdefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionMinimal 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…statement
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.
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  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. Degree of a field extension definition
  27. Minimal polynomial of an algebraic element definition
  28. Algebraic extension definition
  29. Finite extension definition
  30. Separable extension definition
  31. Normal extension definition
  32. The Galois group has order at most the degree statement
  33. Galois extension definition
  34. Order of the Galois group of a Galois extension equals its degree statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Galois_extension, Characterization of Galois extensions
  2. Web https://en.wikipedia.org/wiki/Galois_group, Cardinality of the Galois group and the degree of the field extension

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