Math 2.0

Field theory and polynomials Definition

Galois group of a polynomial

Builds on Galois groupSplitting field

Definition

Let KK be a field and let f∈K[x]f \in K[x] be nonzero. Let LL be a splitting field of ff over KK. The Galois group of ff over KK is the group Aut⁡(L/K)\operatorname{Aut}(L/K) of the automorphisms of L/KL/K.

Remarks

By the uniqueness of splitting fields, the group does not depend on the choice of LL, up to isomorphism. Each automorphism permutes the distinct roots r1,…,rmr_1, \dots, r_m of ff in LL, and it is determined by this permutation, because the roots generate LL. So the group is isomorphic to a subgroup of the symmetric group SmS_m. The Galois group of x2−2x^2 - 2 over ℚ\mathbb{Q} has order 22. The Galois group of x3−2x^3 - 2 over ℚ\mathbb{Q} is isomorphic to S3S_3. If ff is separable, then L/KL/K is a Galois extension.

Reading path

To understand this page from the ground up, read these 19 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 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 ringdefinitionPolynomial evaluation, definition, no evidence levelPolynomial evaluationdefinitionField, definition, no evidence levelFielddefinitionRing isomorphism, definition, no evidence levelRing isomorphismdefinitionRoot of a polynomial, definition, no evidence levelRoot of a polynomialdefinitionSubfield, definition, no evidence levelSubfielddefinitionSplit polynomial, definition, no evidence levelSplit polynomialdefinitionField extension, definition, no evidence levelField extensiondefinitionGalois group, definition, no evidence levelGalois groupdefinitionIntermediate field, definition, no evidence levelIntermediate fielddefinitionField generated by adjoining elements, definition, no evidence levelField generated by ad…definitionSplitting field, definition, no evidence levelSplitting fielddefinitionGalois group of a polynomial, definition, no evidence levelGalois group of a pol…definition
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. Polynomial ring definition
  5. Ring homomorphism definition
  6. Subring definition
  7. Unit of a ring definition
  8. Polynomial evaluation definition
  9. Field definition
  10. Ring isomorphism definition
  11. Root of a polynomial definition
  12. Subfield definition
  13. Split polynomial definition
  14. Field extension definition
  15. Galois group definition
  16. Intermediate field definition
  17. Field generated by adjoining elements definition
  18. Splitting field definition
  19. Galois group of a polynomial definition

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Galois_group, Galois group of a polynomial
About this page
Tag0JZY3R908H
URIm2://0JZY3R908H
Kinddefinition
WritingW0 stub
MSC12F10
DefinesGalois group of a polynomial
UsesGalois group, Splitting field

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

Results about this

Used by