Math 2.0

Field theory and polynomials Theorem

The Galois group has order at most the degree

Builds on Galois groupFinite extension

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.

Reading path

To understand this page from the ground up, read these 14 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 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
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. 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. 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

Where this leads

3 pages build on this one.

Further reading

  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
About this page
Tag0QM98DNWVW
URIm2://0QM98DNWVW
Kindstatement, theorem
WritingW0 stub
MSC12F10
StatusThis copy of the site has no build data. Run m2 status 0QM98DNWVW.
UsesGalois group, Finite extension

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (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