Math 2.0

Field theory and polynomials Field extensions Theorem

Uniqueness of splitting fields

Builds on Splitting fieldK(a) is isomorphic to K[x] modulo the minimal polynomial

Statement

Let KK be a field, let f∈K[x]f \in K[x] be nonzero, and let LL and L′L' be splitting fields of ff over KK. Then there 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.

Remarks

The proof is by induction on [L:K][L : K]. It extends an isomorphism one root at a time, with the structure of simple extensions. The isomorphism σ\sigma is usually not unique. The isomorphisms from LL to itself that fix KK form the Galois group of ff. So one speaks of "the" splitting field of ff over KK. The choice of the irreducible factors in the construction does not change the result, up to isomorphism.

Reading path

To understand this page from the ground up, read these 52 pages in order. The foundations come first.

  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Ideal definition
  5. Polynomial ring definition
  6. Ring homomorphism definition
  7. Subring definition
  8. Unit of a ring definition
  9. Zero divisor definition
  10. Divisibility in a commutative ring definition
  11. Ideal generated by a subset definition
  12. Maximal ideal definition
  13. Two-sided ideal definition
  14. Degree of a polynomial definition
  15. Polynomial evaluation definition
  16. Field definition
  17. Integral domain definition
  18. Kernel of a ring homomorphism definition
  19. Ring isomorphism definition
  20. Principal ideal definition
  21. Quotient ring definition
  22. The only ideals of a field are zero and the whole field statement
  23. Evaluation homomorphism definition
  24. Leading coefficient definition
  25. Root of a polynomial definition
  26. Subfield definition
  27. Irreducible element definition
  28. Euclidean domain definition
  29. Correspondence theorem for rings statement
  30. Principal ideal domain definition
  31. Split polynomial definition
  32. Monic polynomial definition
  33. Universal property of the quotient ring statement
  34. Irreducible polynomial definition
  35. Degree of a product statement
  36. Field extension definition
  37. R/M is a field iff M is maximal statement
  38. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  39. Division algorithm for polynomials statement
  40. Algebraic element definition
  41. R[x] is an integral domain iff R is statement
  42. First isomorphism theorem for rings statement
  43. Intermediate field definition
  44. Minimal polynomial of an algebraic element definition
  45. K[x] is a Euclidean domain statement
  46. Field generated by adjoining elements definition
  47. The minimal polynomial divides every polynomial vanishing at the element statement
  48. K[x]/(f) is a field iff f is irreducible statement
  49. Simple extension definition
  50. Splitting field definition
  51. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  52. Uniqueness of splitting fields statement

Where this leads

8 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Splitting_field, Constructing splitting fields
About this page
Tag069A3BC3EP
URIm2://069A3BC3EP
Kindstatement, theorem
WritingW0 stub
MSC12F05
StatusThis copy of the site has no build data. Run m2 status 069A3BC3EP.
UsesSplitting field, K(a) is isomorphic to K[x] modulo the minimal polynomial

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