Math 2.0

Field theory and polynomials Definition

Galois group

Builds on Field extensionRing isomorphismGroup

Definition

Let L/KL/K be a field extension. An automorphism of L/KL/K is a ring isomorphism σ ⁣:L→L\sigma \colon L \to L with σ(c)=c\sigma(c) = c for every c∈Kc \in K. These automorphisms form a group Aut⁡(L/K)\operatorname{Aut}(L/K) under composition. If L/KL/K is a Galois extension, this group is the Galois group of L/KL/K, written Gal⁡(L/K)\operatorname{Gal}(L/K).

Remarks

Some authors call Aut⁡(L/K)\operatorname{Aut}(L/K) the Galois group for every extension L/KL/K. The group Gal⁡(ℂ/ℝ)\operatorname{Gal}(\mathbb{C}/\mathbb{R}) has two elements: the identity and complex conjugation. The group Aut⁡(ℚ(23)/ℚ)\operatorname{Aut}(\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}) is trivial, because an automorphism must send 23\sqrt[3]{2} to a real cube root of 22. For a finite extension, ∣Aut⁡(L/K)∣≤[L:K]|\operatorname{Aut}(L/K)| \le [L : K], by the bound on the order.

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Group, concept, no evidence levelGroupconceptRing, definition, no evidence levelRingdefinitionCommutative ring, definition, no evidence levelCommutative ringdefinitionRing homomorphism, definition, no evidence levelRing homomorphismdefinitionSubring, definition, no evidence levelSubringdefinitionUnit of a ring, definition, no evidence levelUnit of a ringdefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionField, definition, no evidence levelFielddefinitionSubfield, definition, no evidence levelSubfielddefinitionField extension, definition, no evidence levelField extensiondefinitionGalois group, definition, no evidence levelGalois groupdefinition
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Ring homomorphism definition
  5. Subring definition
  6. Unit of a ring definition
  7. Ring isomorphism definition
  8. Field definition
  9. Subfield definition
  10. Field extension definition
  11. Galois group definition

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23 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Galois_group, Definition
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DefinesGalois group
UsesField extension, Ring isomorphism, Group

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