Math 2.0

Field theory and polynomials Proof

Proof that the degree of a simple extension is the degree of the minimal polynomial

Builds on Field extensionMinimal polynomial of an algebraic elementK(a) is isomorphic to K[x] modulo the minimal polynomialDivision algorithm for polynomialsThe minimal polynomial divides every polynomial vanishing at the elementDegree of a productDegree of a field extensionBasisSpanning setLinear independenceDimension

Proof

  1. Let L/KL/K be a field extension, let a∈La \in L be algebraic over KK, and let mam_a be its minimal polynomial, of degree nn.
  2. By the structure of simple extensions, K(a)={f(a):f∈K[x]}K(a) = \set{ f(a) : f \in K[x] }.
  3. Spanning. Let z∈K(a)z \in K(a), so z=f(a)z = f(a) for some f∈K[x]f \in K[x] by Step 2.
  4. Since mam_a is monic, polynomial division gives f=qma+rf = q m_a + r with deg⁡r<n\deg r < n. Write r=∑i=0n−1cixir = \sum_{i=0}^{n-1} c_i x^i with ci∈Kc_i \in K.
  5. Then z=f(a)=q(a)ma(a)+r(a)=r(a)=∑i=0n−1ciaiz = f(a) = q(a) m_a(a) + r(a) = r(a) = \sum_{i=0}^{n-1} c_i a^i. So 1,a,…,an−11, a, \dots, a^{n-1} span K(a)K(a) over KK.
  6. Linear independence. Let c0,…,cn−1∈Kc_0, \dots, c_{n-1} \in K with ∑i=0n−1ciai=0\sum_{i=0}^{n-1} c_i a^i = 0, and let r=∑i=0n−1cixir = \sum_{i=0}^{n-1} c_i x^i. Then r(a)=0r(a) = 0.
  7. By divisibility, r=masr = m_a s for some s∈K[x]s \in K[x]. If s≠0s \ne 0, then deg⁡r=n+deg⁡s≥n\deg r = n + \deg s \ge n by the degree of a product, which is false.
  8. So s=0s = 0, hence r=0r = 0, and all ci=0c_i = 0.
  9. By Steps 5 and 8, 1,a,…,an−11, a, \dots, a^{n-1} is a basis of K(a)K(a) over KK.
  10. This basis has nn elements, so by the definition of dimension, [K(a):K]=n[K(a) : K] = n, as a degree.

Idea

Every element of K(a)K(a) is a polynomial in aa, and dividing by mam_a cuts its degree below nn. So the powers 1,a,…,an−11, a, \dots, a^{n-1} span. A linear relation among them would be a nonzero polynomial of degree less than nn with root aa, which the minimality of mam_a forbids.

Reading path

To understand this page from the ground up, read these 117 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 of the universal property of the polynomial ring, Proof that R[x] is an integral domain iff R is, 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 that K(a) is isomorphic to K[x] modulo the minimal polynomial, 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 that K[x]/(f) is a field iff f is irreducible, Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal, 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. Mathematical induction concept
  5. Natural numbers object
  6. Peano axioms concept
  7. Proposition concept
  8. Spanning set concept
  9. Vector space definition
  10. Set definition
  11. Ring definition
  12. Successor function definition
  13. Implication definition
  14. Predicate definition
  15. Linear combination definition
  16. Linear subspace definition
  17. Ordered pair definition
  18. Commutative ring definition
  19. Polynomial ring definition
  20. Ring homomorphism definition
  21. Subring definition
  22. Unit of a ring definition
  23. Zero divisor definition
  24. Ideal definition
  25. Addition of natural numbers definition
  26. The successor function is injective statement
  27. Zero is not a successor statement
  28. Existential quantifier definition
  29. Universal quantifier definition
  30. Linear independence definition
  31. Linear span definition
  32. Cartesian product definition
  33. Divisibility in a commutative ring definition
  34. Degree of a polynomial definition
  35. Polynomial evaluation definition
  36. Proof of the universal property of the polynomial ring proof
  37. Field definition
  38. Kernel of a ring homomorphism definition
  39. Ring isomorphism definition
  40. Integral domain definition
  41. Ideal generated by a subset definition
  42. Subset definition
  43. Proof of the commutativity of addition by induction proof
  44. Order on the natural numbers definition
  45. Maximal ideal definition
  46. Two-sided ideal definition
  47. Uniqueness quantifier definition
  48. Spanning set definition
  49. Leading coefficient definition
  50. Evaluation homomorphism definition
  51. Root of a polynomial definition
  52. Proof of cancellation in an integral domain proof
  53. Subfield definition
  54. Binary relation definition
  55. Proof that the only ideals of a field are zero and the whole field proof
  56. Irreducible element definition
  57. Commutativity of addition statement
  58. Proof of strong induction from ordinary induction proof
  59. Quotient ring definition
  60. Euclidean domain definition
  61. Principal ideal definition
  62. Basis definition
  63. Finite-dimensional vector space definition
  64. Monic polynomial definition
  65. Universal property of the polynomial ring statement
  66. Proof of the degree of a product over an integral domain proof
  67. Field extension definition
  68. Irreducible polynomial definition
  69. Cancellation in an integral domain statement
  70. The only ideals of a field are zero and the whole field statement
  71. Proof of the universal property of the quotient ring proof
  72. Proof that R/M is a field iff M is maximal proof
  73. Strong induction statement
  74. Function definition
  75. Principal ideal domain definition
  76. Dimension definition
  77. Division algorithm for polynomials statement
  78. Algebraic element definition
  79. Degree of a field extension definition
  80. Degree of a product statement
  81. Intermediate field definition
  82. Universal property of the quotient ring statement
  83. Proof of the well-ordering principle from strong induction proof
  84. Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal proof
  85. Codomain definition
  86. Image of a set definition
  87. Injective function definition
  88. Preimage of a set definition
  89. Minimal polynomial of an algebraic element definition
  90. Proof that R[x] is an integral domain iff R is proof
  91. Field generated by adjoining elements definition
  92. Proof that a ring homomorphism is injective iff its kernel is zero proof
  93. Well-ordering principle statement
  94. Surjective function definition
  95. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  96. Proof that the minimal polynomial divides every polynomial vanishing at the element proof
  97. Simple extension definition
  98. Proof that the minimal polynomial is irreducible proof
  99. R[x] is an integral domain iff R is statement
  100. A ring homomorphism is injective iff its kernel is zero statement
  101. Bijective function definition
  102. Proof that Euclidean domains are principal ideal domains proof
  103. The minimal polynomial divides every polynomial vanishing at the element statement
  104. The minimal polynomial is irreducible statement
  105. Proof that K[x] is a Euclidean domain proof
  106. Proof of the first isomorphism theorem for rings proof
  107. Proof of the correspondence theorem for rings proof
  108. Euclidean domains are principal ideal domains statement
  109. K[x] is a Euclidean domain statement
  110. First isomorphism theorem for rings statement
  111. Correspondence theorem for rings statement
  112. R/M is a field iff M is maximal statement
  113. Proof that K[x]/(f) is a field iff f is irreducible proof
  114. K[x]/(f) is a field iff f is irreducible statement
  115. Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial proof
  116. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  117. Proof that the degree of a simple extension is the degree of the minimal polynomial proof

Where this leads

47 pages build on this one.

Further reading

  1. Web https://en.wikipedia.org/wiki/Simple_extension, Definition
  2. Web https://www.jmilne.org/math/CourseNotes/FT.pdf, Chapter 1, Stem fields
About this page
Tag9SB00E1N98
URIm2://9SB00E1N98
Kindproof
WritingW0 stub
MSC12F05
ProvesDegree of a simple extension is the degree of the minimal polynomial
UsesField extension, Minimal polynomial of an algebraic element, K(a) is isomorphic to K[x] modulo the minimal polynomial, Division algorithm for polynomials, The minimal polynomial divides every polynomial vanishing at the element, Degree of a product, Degree of a field extension, Basis, Spanning set, Linear independence, Dimension
Changed4 November 2026

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