Math 2.0

object

Galois field

Description

For each prime power q=pnq = p^n, the Galois field GF(q)\mathrm{GF}(q), also written Fq\mathbb{F}_q, is the finite field with qq elements. It exists, and it is unique up to isomorphism. One model is the splitting field of xq−xx^q - x over Fp=Z/pZ\mathbb{F}_p = \mathbb{Z}/p\mathbb{Z}. Another model is Fp[x]/(f)\mathbb{F}_p[x]/(f) for an irreducible polynomial f∈Fp[x]f \in \mathbb{F}_p[x] of degree nn.

Remarks

The name honours Évariste Galois. The field GF(4)=F2[x]/(x2+x+1)\mathrm{GF}(4) = \mathbb{F}_2[x]/(x^2 + x + 1) has the elements 0,1,α,α+10, 1, \alpha, \alpha + 1, with α2=α+1\alpha^2 = \alpha + 1. The field GF(pm)\mathrm{GF}(p^m) is isomorphic to a subfield of GF(pn)\mathrm{GF}(p^n) if and only if mm divides nn.

Explanations

L1 human

Since there is only one finite field of each order, up to a change of names, one gives it a fixed name: the Galois field with qq elements. Computations in it are done with polynomials over the integers modulo pp.

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What this needs

To read this node, read these 81 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. Formal derivative definition
  17. Group of units of a ring definition
  18. Polynomial evaluation definition
  19. Field definition
  20. Integral domain definition
  21. Kernel of a ring homomorphism definition
  22. Ring isomorphism definition
  23. Principal ideal definition
  24. Quotient ring definition
  25. Associate elements definition
  26. Prime element definition
  27. Leading coefficient definition
  28. The characteristic of an integral domain is zero or prime statement
  29. The only ideals of a field are zero and the whole field statement
  30. Evaluation homomorphism definition
  31. Subfield definition
  32. Root of a polynomial definition
  33. Irreducible element definition
  34. Finite field definition
  35. Field of fractions definition
  36. Euclidean domain definition
  37. Correspondence theorem for rings statement
  38. Greatest common divisor in an integral domain definition
  39. Monic polynomial definition
  40. Principal ideal domain definition
  41. Universal property of the quotient ring statement
  42. Prime subfield definition
  43. Degree of a product statement
  44. Split polynomial definition
  45. Irreducible polynomial definition
  46. Multiplicity of a root definition
  47. Unique factorisation domain definition
  48. Field extension definition
  49. Every element of a finite field of order q satisfies x^q = x statement
  50. Division algorithm for polynomials statement
  51. R/M is a field iff M is maximal statement
  52. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  53. Content of a polynomial definition
  54. R[x] is an integral domain iff R is statement
  55. First isomorphism theorem for rings statement
  56. Algebraic element definition
  57. Repeated roots and the derivative statement
  58. Intermediate field definition
  59. Degree of a field extension definition
  60. Primitive polynomial definition
  61. Minimal polynomial of an algebraic element definition
  62. K[x] is a Euclidean domain statement
  63. The order of a finite field is a prime power statement
  64. Tower law statement
  65. Field generated by adjoining elements definition
  66. The minimal polynomial divides every polynomial vanishing at the element statement
  67. K[x]/(f) is a field iff f is irreducible statement
  68. Product of primitive polynomials is primitive statement
  69. Simple extension definition
  70. Splitting field definition
  71. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  72. Adjoining a root of a polynomial definition
  73. A finite field of order q is the splitting field of x^q - x statement
  74. Gauss's lemma statement
  75. Uniqueness of splitting fields statement
  76. Polynomial ring over a UFD is a UFD statement
  77. Uniqueness of finite fields of a given order statement
  78. Every polynomial has a root in some extension field statement
  79. Existence of splitting fields statement
  80. Existence of finite fields of every prime power order statement
  81. Galois field object

Sources

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

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