Math 2.0

statement theorem

Existence of splitting fields

Statement

Let KK be a field and let f∈K[x]f \in K[x] have degree n≥1n \ge 1. Then ff has a splitting field LL over KK, and [L:K]≤n![L : K] \le n!.

Remarks

The proof is by induction on nn. Some extension K1K_1 contains a root a1a_1 of ff, with [K1:K]≤n[K_1 : K] \le n. Then f=(x−a1)gf = (x - a_1) g in K1[x]K_1[x], and a splitting field of gg over K1K_1 is a splitting field of ff over KK. The bound n!n! follows from the tower law. The bound is reached: the splitting field of x3−2x^3 - 2 over Q\mathbb{Q} has degree 6=3!6 = 3!.

Explanations

L1 human

You build a splitting field one root at a time. Each step adds a root of a remaining factor, and the degree of the step is at most the degree of that factor.

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What this needs

To read this node, read these 55 nodes in this order. The foundations come first, and the list ends with this node.

  1. Group concept
  2. Ring definition
  3. Commutative ring definition
  4. Ideal definition
  5. Polynomial ring definition
  6. Subring definition
  7. Unit of a ring definition
  8. Zero divisor definition
  9. Divisibility in a commutative ring definition
  10. Ideal generated by a subset definition
  11. Maximal ideal definition
  12. Two-sided ideal definition
  13. Degree of a polynomial definition
  14. Field definition
  15. Polynomial evaluation definition
  16. Integral domain definition
  17. Principal ideal definition
  18. Quotient ring definition
  19. Associate elements definition
  20. Prime element definition
  21. The only ideals of a field are zero and the whole field statement
  22. Leading coefficient definition
  23. Subfield definition
  24. Irreducible element definition
  25. Field of fractions definition
  26. Root of a polynomial definition
  27. Euclidean domain definition
  28. Correspondence theorem for rings statement
  29. Greatest common divisor in an integral domain definition
  30. Principal ideal domain definition
  31. Irreducible polynomial definition
  32. Monic polynomial definition
  33. Degree of a product statement
  34. Unique factorisation domain definition
  35. Field extension definition
  36. Split polynomial 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. Content of a polynomial definition
  41. R[x] is an integral domain iff R is statement
  42. Intermediate field definition
  43. Degree of a field extension definition
  44. Primitive polynomial definition
  45. K[x] is a Euclidean domain statement
  46. Tower law statement
  47. Field generated by adjoining elements definition
  48. K[x]/(f) is a field iff f is irreducible statement
  49. Product of primitive polynomials is primitive statement
  50. Splitting field definition
  51. Adjoining a root of a polynomial definition
  52. Gauss's lemma statement
  53. Polynomial ring over a UFD is a UFD statement
  54. Every polynomial has a root in some extension field statement
  55. Existence of splitting fields statement

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Sources

  1. Web https://en.wikipedia.org/wiki/Splitting_field, Constructing splitting fields

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