Math 2.0

statement theorem

The Galois group has order at most the degree

Statement

Let L/KL/K be a finite extension. Then the group Aut⁡(L/K)\operatorname{Aut}(L/K) of automorphisms of L/KL/K is finite, and ∣Aut⁡(L/K)∣≤[L:K].|\operatorname{Aut}(L/K)| \le [L : K].

Remarks

For a simple extension L=K(a)L = K(a), an automorphism is determined by σ(a)\sigma(a), which is a root in LL of the minimal polynomial of aa. There are at most [L:K][L : K] such roots. Equality holds if and only if L/KL/K is a Galois extension. For Q(23)/Q\mathbb{Q}(\sqrt[3]{2})/\mathbb{Q}, the inequality is strict: 1<31 < 3.

Explanations

L1 human

A finite extension has only finitely many symmetries over the base field, and at most as many as its degree. When there are exactly as many as the degree, the extension is Galois.

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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 extensiondefinitionDegree of a field extension, definition, no evidence levelDegree of a field ext…definitionGalois group, definition, no evidence levelGalois groupdefinitionFinite extension, definition, no evidence levelFinite extensiondefinitionThe Galois group has order at most the degree, statement, no evidence levelThe Galois group has …statement
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  1. Group concept
  2. Ring definition
  3. Commutative ring definition
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  5. Subring definition
  6. Unit of a ring definition
  7. Ring isomorphism definition
  8. Field definition
  9. Subfield definition
  10. Field extension definition
  11. Degree of a field extension definition
  12. Galois group definition
  13. Finite extension definition
  14. The Galois group has order at most the degree statement

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Sources

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

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