Math 2.0

statement theorem

Uniqueness of finite fields of a given order

Statement

Let FF and F′F' be finite fields with ∣F∣=∣F′∣|F| = |F'|. Then FF and F′F' are isomorphic.

Remarks

The proof: let q=∣F∣q = |F|. Both fields have characteristic pp and prime subfield Fp\mathbb{F}_p. Both are splitting fields of xq−xx^q - x over Fp\mathbb{F}_p, so they are isomorphic by the uniqueness of splitting fields. Also, a field contains at most one subfield of order qq: its elements must be the qq roots of xq−xx^q - x.

History

E. H. Moore first proved the classification of finite fields in 1893.

Explanations

L1 human

Two finite fields with the same number of elements have the same structure. Only the names of the elements can differ.

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What this needs

To read this node, read these 62 nodes in this order. The foundations come first, and the list ends with this node.

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

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Sources

  1. Web https://en.wikipedia.org/wiki/Finite_field, Existence and uniqueness

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