Math 2.0

Field theory and polynomials Proof

Proof of the existence of finite fields of every prime power order

Builds on Existence of splitting fieldsSplitting fieldSplit polynomialFormal derivativeRepeated roots and the derivativeMultiplicity of a rootCharacteristic of a ringFrobenius endomorphismFreshman's dreamSubfieldFinite field

Proof

  1. Let pp be prime, let n≥1n \ge 1 and let q=pnq = p^n. Let f=xq−x∈𝔽p[x]f = x^q - x \in \F_p[x].
  2. By the existence of splitting fields, ff has a splitting field LL over 𝔽p\F_p. So f=(x−a1)⋯(x−aq)f = (x - a_1) \cdots (x - a_q) with ai∈La_i \in L, since ff is monic of degree qq and splits over LL.
  3. The field LL contains 𝔽p\F_p, so p⋅1=0p \cdot 1 = 0 in LL, and the characteristic of LL is pp.
  4. The formal derivative of ff is f′=qxq−1−1f' = q x^{q-1} - 1. Since q⋅1=pn−1(p⋅1)=0q \cdot 1 = p^{n-1} (p \cdot 1) = 0 in LL, we get f′=−1f' = -1.
  5. Suppose that some root cc of ff in LL has multiplicity at least 22. Then f′(c)=0f'(c) = 0 by the criterion for repeated roots. This contradicts f′(c)=−1≠0f'(c) = -1 \neq 0.
  6. So every root of ff has multiplicity 11, and a1,…,aqa_1, \dots, a_q are distinct. Let R={a1,…,aq}R = \set{a_1, \dots, a_q}, so ∣R∣=q\abs{R} = q and R={a∈L:aq=a}R = \set{a \in L : a^q = a}.
  7. The Frobenius endomorphism φ(a)=ap\varphi(a) = a^p of LL is a ring homomorphism, by the freshman's dream. So ψ=φn\psi = \varphi^n, ψ(a)=aq\psi(a) = a^{q}, is a ring homomorphism L→LL \to L.
  8. So R={a∈L:ψ(a)=a}R = \set{a \in L : \psi(a) = a} contains 00 and 11. For a,b∈Ra, b \in R, we get ψ(a−b)=ψ(a)−ψ(b)=a−b\psi(a - b) = \psi(a) - \psi(b) = a - b and ψ(ab)=ψ(a)ψ(b)=ab\psi(ab) = \psi(a)\psi(b) = ab.
  9. For a∈Ra \in R with a≠0a \neq 0, we get ψ(a−1)=ψ(a)−1=a−1\psi(a^{-1}) = \psi(a)^{-1} = a^{-1}, since ψ(a)ψ(a−1)=ψ(1)=1\psi(a)\psi(a^{-1}) = \psi(1) = 1.
  10. By steps 8 and 9, RR is a subfield of LL. By step 6 it has q=pnq = p^n elements, so it is a finite field of order pnp^n.

Idea

Build a field in which xq−xx^q - x splits, and look at its roots. The derivative is the constant −1-1, so the roots are distinct and there are exactly qq of them. The map a↦aqa \mapsto a^q is a ring homomorphism in characteristic pp, so its fixed points form a field.

Reading path

To understand this page from the ground up, read these 202 pages in order. The foundations come first. The path takes the proofs Proof of Bézout's identity, Proof that in a UFD irreducible elements are prime, Proof of cancellation in an integral domain, Proof of the generalized Euclid's lemma, Proof that ring homomorphisms from a field are injective, Proof of the commutativity of addition by induction, Proof of the universal property of the polynomial ring, Proof of the tower law, Proof of Euclid's lemma, Proof that every polynomial has a root in some extension field, Proof that R[x] is an integral domain iff R is, Proof that the degree of a simple extension is the degree of the minimal polynomial, Proof that the product of primitive polynomials is primitive, Proof of the first isomorphism theorem for rings, Proof that a ring homomorphism is injective iff its kernel is zero, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof of the correspondence theorem for rings, Proof of Pascal's rule from the recursive definition, Proof that R/M is a field iff M is maximal, Proof that the only ideals of a field are zero and the whole field, Proof of the remainder theorem, Proof of the freshman's dream, Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial, Proof of the factorial formula for binomial coefficients by induction, Proof of the existence of splitting fields, Proof of Zorn's lemma, Proof of the binomial theorem by induction, Proof of the well-ordering principle from strong induction, Proof that the minimal polynomial is irreducible, Proof of the division algorithm, Proof that the minimal polynomial divides every polynomial vanishing at the element, Proof of Gauss's lemma on irreducibility over a UFD, Proof that K[x]/(f) is a field iff f is irreducible, Proof of the derivative test for repeated roots, Proof that prime elements are irreducible in an integral domain, Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal, Proof of the factor theorem, Proof that every vector space has a basis, by Zorn's lemma, Proof that Euclidean domains are principal ideal domains, Proof of the universal property of the quotient ring, Proof that K[x] is a Euclidean domain.

  1. Binomial coefficient definition
  2. Definition by recursion concept
  3. Divisibility definition
  4. Element definition
  5. Factorial definition
  6. Group concept
  7. Linear combination concept
  8. Mathematical induction concept
  9. Natural numbers object
  10. Peano axioms concept
  11. Proof of Zorn's lemma proof
  12. Proposition concept
  13. Spanning set concept
  14. Vector space definition
  15. Proof of Pascal's rule from the recursive definition proof
  16. Greatest common divisor definition
  17. Prime number definition
  18. Set definition
  19. Ring definition
  20. Successor function definition
  21. Disjunction definition
  22. Implication definition
  23. Predicate definition
  24. Linear combination definition
  25. Linear subspace definition
  26. Pascal's rule statement
  27. Coprime integers definition
  28. Family of sets definition
  29. Ordered pair definition
  30. Characteristic of a ring definition
  31. Commutative ring definition
  32. Polynomial ring definition
  33. Ring homomorphism definition
  34. Subring definition
  35. Union definition
  36. Unit of a ring definition
  37. Zero divisor definition
  38. Ideal definition
  39. Addition of natural numbers definition
  40. The successor function is injective statement
  41. Zero is not a successor statement
  42. Existential quantifier definition
  43. Universal quantifier definition
  44. Linear independence definition
  45. Linear span definition
  46. Proof of the factorial formula for binomial coefficients by induction proof
  47. Cartesian product definition
  48. Divisibility in a commutative ring definition
  49. Degree of a polynomial definition
  50. Formal derivative definition
  51. Polynomial evaluation definition
  52. Proof of the universal property of the polynomial ring proof
  53. Kernel of a ring homomorphism definition
  54. Ring isomorphism definition
  55. Field definition
  56. Subset definition
  57. Integral domain definition
  58. Proof of the commutativity of addition by induction proof
  59. Ideal generated by a subset definition
  60. Union of a family of sets definition
  61. Order on the natural numbers definition
  62. Maximal ideal definition
  63. Two-sided ideal definition
  64. Uniqueness quantifier definition
  65. Spanning set definition
  66. Factorial formula for binomial coefficients statement
  67. Leading coefficient definition
  68. Root of a polynomial definition
  69. Binary relation definition
  70. Associate elements definition
  71. Evaluation homomorphism definition
  72. Prime element definition
  73. Proof of cancellation in an integral domain proof
  74. Subfield definition
  75. Finite field definition
  76. Field of fractions definition
  77. Irreducible element definition
  78. Proof that the only ideals of a field are zero and the whole field proof
  79. Quotient ring definition
  80. Commutativity of addition statement
  81. Euclidean domain definition
  82. Proof of strong induction from ordinary induction proof
  83. Principal ideal definition
  84. Basis definition
  85. Finite-dimensional vector space definition
  86. Monic polynomial definition
  87. Proof of the binomial theorem by induction proof
  88. Multiplicity of a root definition
  89. Antisymmetric relation definition
  90. Reflexive relation definition
  91. Transitive relation definition
  92. Split polynomial definition
  93. Proof of the remainder theorem proof
  94. Proof of the degree of a product over an integral domain proof
  95. Greatest common divisor in an integral domain definition
  96. Universal property of the polynomial ring statement
  97. Field extension definition
  98. Irreducible polynomial definition
  99. Unique factorisation domain definition
  100. Cancellation in an integral domain statement
  101. Proof of the universal property of the quotient ring proof
  102. Function definition
  103. Proof that R/M is a field iff M is maximal proof
  104. The only ideals of a field are zero and the whole field statement
  105. Strong induction statement
  106. Principal ideal domain definition
  107. Dimension theorem for vector spaces statement
  108. Dimension definition
  109. Binomial theorem statement
  110. Division algorithm for polynomials statement
  111. Preorder definition
  112. Degree of a product statement
  113. Algebraic element definition
  114. Proof that prime elements are irreducible in an integral domain proof
  115. Intermediate field definition
  116. Universal property of the quotient ring statement
  117. Content of a polynomial definition
  118. Degree of a field extension definition
  119. Proof of the well-ordering principle from strong induction proof
  120. Image of a set definition
  121. Preimage of a set definition
  122. Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal proof
  123. Choice function definition
  124. Codomain definition
  125. Injective function definition
  126. Minimal polynomial of an algebraic element definition
  127. Partial order definition
  128. Remainder theorem statement
  129. Proof that R[x] is an integral domain iff R is proof
  130. Primitive polynomial definition
  131. Prime elements are irreducible in an integral domain statement
  132. Field generated by adjoining elements definition
  133. Proof that a ring homomorphism is injective iff its kernel is zero proof
  134. Well-ordering principle statement
  135. Axiom of choice statement
  136. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  137. Surjective function definition
  138. Maximal element definition
  139. Total order definition
  140. Upper bound definition
  141. Proof of the factor theorem proof
  142. Proof that the minimal polynomial divides every polynomial vanishing at the element proof
  143. R[x] is an integral domain iff R is statement
  144. Proof that the minimal polynomial is irreducible proof
  145. Simple extension definition
  146. A ring homomorphism is injective iff its kernel is zero statement
  147. Splitting field definition
  148. Proof that in a UFD irreducible elements are prime proof
  149. Proof of the division algorithm proof
  150. Bijective function definition
  151. Proof that Euclidean domains are principal ideal domains proof
  152. Chain definition
  153. The minimal polynomial divides every polynomial vanishing at the element statement
  154. Factor theorem statement
  155. Proof that K[x] is a Euclidean domain proof
  156. The minimal polynomial is irreducible statement
  157. Division algorithm statement
  158. Proof that ring homomorphisms from a field are injective proof
  159. In a UFD irreducible elements are prime statement
  160. Proof of the first isomorphism theorem for rings proof
  161. Proof of the correspondence theorem for rings proof
  162. Equinumerous sets definition
  163. Euclidean domains are principal ideal domains statement
  164. Proof of the derivative test for repeated roots proof
  165. Proof of Bézout's identity proof
  166. Zorn's lemma statement
  167. K[x] is a Euclidean domain statement
  168. Ring homomorphisms from a field are injective statement
  169. Proof that the product of primitive polynomials is primitive proof
  170. Proof of Gauss's lemma on irreducibility over a UFD proof
  171. First isomorphism theorem for rings statement
  172. Cardinality definition
  173. Correspondence theorem for rings statement
  174. Bézout's identity statement
  175. Repeated roots and the derivative statement
  176. Product of primitive polynomials is primitive statement
  177. Proof that every vector space has a basis, by Zorn's lemma proof
  178. R/M is a field iff M is maximal statement
  179. Proof of the generalized Euclid's lemma proof
  180. Gauss's lemma statement
  181. Proof that K[x]/(f) is a field iff f is irreducible proof
  182. Every vector space has a basis statement
  183. Generalized Euclid's lemma statement
  184. K[x]/(f) is a field iff f is irreducible statement
  185. Polynomial ring over a UFD is a UFD statement
  186. Proof of the tower law proof
  187. Proof of Euclid's lemma proof
  188. Tower law statement
  189. Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial proof
  190. Adjoining a root of a polynomial definition
  191. Euclid's lemma statement
  192. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  193. Proof of the freshman's dream proof
  194. Proof that the degree of a simple extension is the degree of the minimal polynomial proof
  195. Degree of a simple extension is the degree of the minimal polynomial statement
  196. Freshman's dream statement
  197. Frobenius endomorphism definition
  198. Proof that every polynomial has a root in some extension field proof
  199. Every polynomial has a root in some extension field statement
  200. Proof of the existence of splitting fields proof
  201. Existence of splitting fields statement
  202. Proof of the existence of finite fields of every prime power order proof

Where this leads

4 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Finite_field, Existence and uniqueness
About this page
TagPV1H86FB4Y
URIm2://PV1H86FB4Y
Kindproof
WritingW0 stub
MSC12E20
ProvesExistence of finite fields of every prime power order
UsesExistence of splitting fields, Splitting field, Split polynomial, Formal derivative, Repeated roots and the derivative, Multiplicity of a root, Characteristic of a ring, Frobenius endomorphism, Freshman's dream, Subfield, Finite field
Changed4 November 2026

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