Math 2.0

Field theory and polynomials Proof

Proof that a finite field of order q is the splitting field of x^q - x

Builds on Finite fieldThe order of a finite field is a prime powerPrime subfieldEvery element of a finite field of order q satisfies x^q = xSplit polynomialField generated by adjoining elementsSplitting field

Proof

  1. Let FF be a finite field of order q=pnq = p^n, with pp prime. By the prime power theorem, q=p′n′q = p'^{n'}, where p′p' is the characteristic of FF. By unique factorisation of qq, p′=pp' = p.
  2. Let PP be the prime subfield of FF. It is the copy 𝔽p\F_p of the multiples of 11, as in that theorem.
  3. The polynomial f=xq−xf = x^q - x has coefficients 11, −1-1 and 00, so f∈P[x]f \in P[x], and F/PF/P is a field extension.
  4. By the factorisation of xq−xx^q - x, f=∏a∈F(x−a)f = \prod_{a \in F} (x - a) in F[x]F[x]. So ff splits over FF, and its roots are exactly the elements of FF.
  5. The field generated over PP by the roots of ff is P(F)P(F). This is the smallest subfield of FF that contains FF, so it is FF.
  6. By steps 4 and 5, FF satisfies both conditions of a splitting field of xq−xx^q - x over 𝔽p\F_p.

Idea

Every element of FF is a root of xq−xx^q - x, and this polynomial has no other roots. So it splits over FF, and its roots already make up the whole field.

Reading path

To understand this page from the ground up, read these 151 pages in order. The foundations come first. The path takes the proofs Proof of cancellation in an integral domain, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that g to the n equals 1 iff the order divides n, Proof that R[x] is an integral domain iff R is, Proof that g to the order of the group is the identity, Proof of the first isomorphism theorem for rings, Proof that a ring homomorphism is injective iff its kernel is zero, Proof that the order of an element divides the order of the group, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof that the characteristic of an integral domain is zero or prime, Proof that a polynomial over a field has at most as many roots as its degree, Proof of the remainder theorem, Proof that the order of a finite field is a prime power, Proof that the order of a cyclic subgroup equals the order of its generator, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the uniqueness of inverses, Proof of Lagrange's theorem, Proof that every element of a finite field of order q satisfies x^q = x, Proof of the factor theorem, Proof that cosets have the same cardinality as the subgroup, Proof of the universal property of the quotient ring.

  1. Binary operation definition
  2. Definition by recursion concept
  3. Divisibility definition
  4. Element definition
  5. Group concept
  6. Left coset concept
  7. Mathematical induction concept
  8. Natural numbers object
  9. Peano axioms concept
  10. Proposition concept
  11. Vector space definition
  12. Identity element definition
  13. Prime number definition
  14. Set definition
  15. Order of a group definition
  16. Ring definition
  17. Subgroup definition
  18. Successor function definition
  19. Conjunction definition
  20. Disjunction definition
  21. Implication definition
  22. Predicate definition
  23. Linear combination definition
  24. Linear subspace definition
  25. Inverse element definition
  26. Composite number definition
  27. Empty set object
  28. Family of sets definition
  29. Ordered pair definition
  30. Characteristic of a ring definition
  31. Commutative ring definition
  32. Ideal definition
  33. Intersection definition
  34. Polynomial ring definition
  35. Ring homomorphism definition
  36. Subring definition
  37. Unit of a ring definition
  38. Zero divisor definition
  39. Union definition
  40. Addition of natural numbers definition
  41. Left coset definition
  42. Right coset definition
  43. Set-builder notation concept
  44. The successor function is injective statement
  45. Zero is not a successor statement
  46. Biconditional definition
  47. Existential quantifier definition
  48. Universal quantifier definition
  49. Linear independence definition
  50. Linear span definition
  51. Proof of the uniqueness of inverses proof
  52. Cartesian product definition
  53. Disjoint sets definition
  54. Divisibility in a commutative ring definition
  55. Two-sided ideal definition
  56. Group of units of a ring definition
  57. Degree of a polynomial definition
  58. Field definition
  59. Polynomial evaluation definition
  60. Proof of the commutativity of addition by induction proof
  61. Subset definition
  62. Integral domain definition
  63. Kernel of a ring homomorphism definition
  64. Ring isomorphism definition
  65. Order on the natural numbers definition
  66. Axiom of extensionality statement
  67. Union of a family of sets definition
  68. Uniqueness quantifier definition
  69. Spanning set definition
  70. Uniqueness of inverses statement
  71. Quotient ring definition
  72. Proof that the characteristic of an integral domain is zero or prime proof
  73. Binary relation definition
  74. Leading coefficient definition
  75. Finite field definition
  76. Proof of cancellation in an integral domain proof
  77. Root of a polynomial definition
  78. Evaluation homomorphism definition
  79. Subfield definition
  80. Partition of a set definition
  81. Commutativity of addition statement
  82. Proof of strong induction from ordinary induction proof
  83. Basis definition
  84. Proof of the laws of exponents in a group proof
  85. Reflexive relation definition
  86. Symmetric relation definition
  87. Transitive relation definition
  88. Monic polynomial definition
  89. The characteristic of an integral domain is zero or prime statement
  90. Proof of the universal property of the quotient ring proof
  91. Multiplicity of a root definition
  92. Prime subfield definition
  93. Proof of the remainder theorem proof
  94. Proof of the degree of a product over an integral domain proof
  95. Field extension definition
  96. Split polynomial definition
  97. Cancellation in an integral domain statement
  98. Function definition
  99. Strong induction statement
  100. Equivalence relation definition
  101. Laws of exponents in a group statement
  102. Division algorithm for polynomials statement
  103. Universal property of the quotient ring statement
  104. Degree of a product statement
  105. Intermediate field definition
  106. Degree of a field extension definition
  107. Proof of the well-ordering principle from strong induction proof
  108. Codomain definition
  109. Injective function definition
  110. Order of an element definition
  111. Cyclic subgroup generated by an element definition
  112. Remainder theorem statement
  113. Proof that R[x] is an integral domain iff R is proof
  114. Field generated by adjoining elements definition
  115. Equivalence class definition
  116. Well-ordering principle statement
  117. Proof that a ring homomorphism is injective iff its kernel is zero proof
  118. Surjective function definition
  119. Proof of the factor theorem proof
  120. Proof of the division algorithm proof
  121. Proof that equivalence classes form a partition proof
  122. R[x] is an integral domain iff R is statement
  123. Bijective function definition
  124. Splitting field definition
  125. A ring homomorphism is injective iff its kernel is zero statement
  126. Division algorithm statement
  127. Factor theorem statement
  128. Equinumerous sets definition
  129. Equivalence classes form a partition statement
  130. Proof of the first isomorphism theorem for rings proof
  131. Proof that g to the n equals 1 iff the order divides n proof
  132. Cardinality definition
  133. First isomorphism theorem for rings statement
  134. Proof that a polynomial over a field has at most as many roots as its degree proof
  135. g to the n equals 1 iff the order divides n statement
  136. A polynomial over a field has at most as many roots as its degree statement
  137. Proof that the order of a finite field is a prime power proof
  138. Proof that cosets have the same cardinality as the subgroup proof
  139. Proof that the order of a cyclic subgroup equals the order of its generator proof
  140. The order of a finite field is a prime power statement
  141. Cosets have the same cardinality as the subgroup statement
  142. Order of a cyclic subgroup equals the order of its generator statement
  143. Proof of Lagrange's theorem proof
  144. Lagrange's theorem statement
  145. Proof that the order of an element divides the order of the group proof
  146. Order of an element divides the order of the group statement
  147. Proof that g to the order of the group is the identity proof
  148. g to the order of the group is the identity statement
  149. Proof that every element of a finite field of order q satisfies x^q = x proof
  150. Every element of a finite field of order q satisfies x^q = x statement
  151. Proof that a finite field of order q is the splitting field of x^q - x proof

Where this leads

5 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_field, Existence and uniqueness
About this page
TagSAMGPXF14S
URIm2://SAMGPXF14S
Kindproof
WritingW0 stub
MSC12E20
ProvesA finite field of order q is the splitting field of x^q - x
UsesFinite field, The order of a finite field is a prime power, Prime subfield, Every element of a finite field of order q satisfies x^q = x, Split polynomial, Field generated by adjoining elements, Splitting field
Changed4 November 2026

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