Math 2.0

Field theory and polynomials Proof

Proof of the existence of splitting fields

Builds on FieldSplitting fieldSplit polynomialField generated by adjoining elementsEvery polynomial has a root in some extension fieldK(a) is isomorphic to K[x] modulo the minimal polynomialTower lawFactor theoremDegree of a productMathematical induction

Proof

We argue by induction on nn. The claim for nn is as follows. For every field KK and every f∈K[x]f \in K[x] of degree nn, there is a splitting field LL of ff over KK with [L:K]≤n![L : K] \le n!.

  1. Case n=1n = 1. Write f=cx+df = c x + d with c≠0c \ne 0, and let a1=−d/c∈Ka_1 = -d/c \in K. Then f=c(x−a1)f = c (x - a_1), so ff splits over KK.
  2. Since a1∈Ka_1 \in K, the generated field K(a1)K(a_1) is KK. So L=KL = K is a splitting field of ff, and [L:K]=1=1![L : K] = 1 = 1!.
  3. Case n≥2n \ge 2. Assume the claim for n−1n - 1, and let f∈K[x]f \in K[x] have degree nn.
  4. There is an extension K′/KK'/K with [K′:K]≤n[K' : K] \le n and a root a1∈K′a_1 \in K' of ff. Let K1=K(a1)⊆K′K_1 = K(a_1) \subseteq K'.
  5. By the tower law, [K′:K]=[K′:K1]⋅[K1:K][K' : K] = [K' : K_1] \cdot [K_1 : K], so [K1:K]≤[K′:K]≤n[K_1 : K] \le [K' : K] \le n.
  6. By the factor theorem in K1[x]K_1[x], f=(x−a1)gf = (x - a_1) g with g∈K1[x]g \in K_1[x]. By the degree of a product, deg⁡g=n−1\deg g = n - 1.
  7. By the induction hypothesis, gg has a splitting field LL over K1K_1 with [L:K1]≤(n−1)![L : K_1] \le (n-1)!. So g=c(x−a2)⋯(x−an)g = c (x - a_2) \cdots (x - a_n) with c,ai∈Lc, a_i \in L, and L=K1(a2,…,an)L = K_1(a_2, \dots, a_n).
  8. Then f=c(x−a1)(x−a2)⋯(x−an)f = c (x - a_1)(x - a_2) \cdots (x - a_n) in L[x]L[x], so ff splits over LL, with roots a1,…,ana_1, \dots, a_n.
  9. Let FF be the subfield of LL generated by KK and a1,…,ana_1, \dots, a_n. We show F=LF = L. Here F⊆LF \subseteq L because FF is a subfield of LL.
  10. By the structure of simple extensions, K1={h(a1):h∈K[x]}K_1 = \set{ h(a_1) : h \in K[x] }. These elements lie in FF, so K1⊆FK_1 \subseteq F.
  11. So FF is a subfield of LL that contains K1K_1 and a2,…,ana_2, \dots, a_n. Since L=K1(a2,…,an)L = K_1(a_2, \dots, a_n) is the smallest such subfield, L⊆FL \subseteq F.
  12. So L=K(a1,…,an)L = K(a_1, \dots, a_n), and with Step 8, LL is a splitting field of ff over KK.
  13. By the tower law and Steps 5 and 7, [L:K]=[L:K1]⋅[K1:K]≤(n−1)!⋅n=n![L : K] = [L : K_1] \cdot [K_1 : K] \le (n-1)! \cdot n = n!.
  14. This completes the induction.

Idea

Adjoin one root of ff, which costs a factor of at most nn in the degree. Over the bigger field, ff is x−a1x - a_1 times a polynomial of degree n−1n - 1, and induction handles that polynomial. The degrees multiply by the tower law, giving at most n⋅(n−1)!=n!n \cdot (n-1)! = n!.

Reading path

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

Where this leads

14 pages build on this one.

Further reading

  1. Web https://www.jmilne.org/math/CourseNotes/FT.pdf, Proposition 2.7
  2. Web https://en.wikipedia.org/wiki/Splitting_field, Constructing splitting fields
About this page
TagND2TPP8Q2W
URIm2://ND2TPP8Q2W
Kindproof
WritingW0 stub
MSC12F05
ProvesExistence of splitting fields
UsesField, Splitting field, Split polynomial, Field generated by adjoining elements, Every polynomial has a root in some extension field, K(a) is isomorphic to K[x] modulo the minimal polynomial, Tower law, Factor theorem, Degree of a product, Mathematical induction
MethodsMathematical induction
Changed4 November 2026

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