Math 2.0

Field theory and polynomials Proof

Proof that an algebraic extension of an algebraic extension is algebraic

Builds on Algebraic extensionAlgebraic elementMinimal polynomial of an algebraic elementDegree of a simple extension is the degree of the minimal polynomialTower lawFinite extensions are algebraicFinite extensionField generated by adjoining elements

Proof

  1. Let K⊆L⊆MK \subseteq L \subseteq M be fields with M/LM/L and L/KL/K algebraic. Let a∈Ma \in M. We show that aa is algebraic over KK.
  2. Since aa is algebraic over LL, it has a minimal polynomial m=xn+cn−1xn−1+⋯+c0m = x^n + c_{n-1} x^{n-1} + \dots + c_0 over LL, with ci∈Lc_i \in L.
  3. Let E0=KE_0 = K, and for 1≤i≤n1 \le i \le n let Ei=Ei−1(ci−1)E_i = E_{i-1}(c_{i-1}), the subfield of MM generated by Ei−1E_{i-1} and ci−1c_{i-1}. Then K=E0⊆E1⊆⋯⊆En⊆MK = E_0 \subseteq E_1 \subseteq \dots \subseteq E_n \subseteq M.
  4. Each ci−1c_{i-1} is algebraic over KK, since L/KL/K is algebraic. A nonzero polynomial over KK with root ci−1c_{i-1} is also one over Ei−1E_{i-1}. So ci−1c_{i-1} is algebraic over Ei−1E_{i-1}.
  5. By the degree of a simple extension, [Ei:Ei−1][E_i : E_{i-1}] equals the degree of the minimal polynomial of ci−1c_{i-1} over Ei−1E_{i-1}. So it is finite.
  6. All coefficients of mm lie in EnE_n, so m∈En[x]m \in E_n[x] is nonzero with m(a)=0m(a) = 0. So aa is algebraic over EnE_n, and [En(a):En][E_n(a) : E_n] is finite by the same result.
  7. By the tower law, applied nn times, [En(a):K]=[En(a):En]⋅[En:En−1]⋯[E1:E0][E_n(a) : K] = [E_n(a) : E_n] \cdot [E_n : E_{n-1}] \cdots [E_1 : E_0]. This is a product of finitely many finite numbers, so it is finite.
  8. So En(a)/KE_n(a)/K is a finite extension. Since finite extensions are algebraic, the element a∈En(a)a \in E_n(a) is algebraic over KK.
  9. Since a∈Ma \in M was arbitrary, M/KM/K is algebraic.

Idea

The element aa satisfies a polynomial with only finitely many coefficients from LL. Adjoining these coefficients to KK gives a finite extension, and aa is algebraic over it. The tower law then makes the whole extension containing aa finite over KK, and finite extensions are algebraic.

Reading path

To understand this page from the ground up, read these 150 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 of the tower law, Proof that R[x] is an integral domain iff R is, Proof that finite extensions are algebraic, Proof that the degree of a simple extension is the degree of the minimal polynomial, 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 every linearly independent set extends to a basis, by Zorn's lemma, 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 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 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. Evaluation homomorphism definition
  58. Proof of cancellation in an integral domain proof
  59. Binary relation definition
  60. Subfield definition
  61. Irreducible element definition
  62. Proof that the only ideals of a field are zero and the whole field proof
  63. Commutativity of addition statement
  64. Euclidean domain definition
  65. Quotient ring definition
  66. Proof of strong induction from ordinary induction proof
  67. Principal ideal definition
  68. Basis definition
  69. Finite-dimensional vector space definition
  70. Monic polynomial definition
  71. Proof of the degree of a product over an integral domain proof
  72. Universal property of the polynomial ring statement
  73. Antisymmetric relation definition
  74. Reflexive relation definition
  75. Transitive relation definition
  76. Irreducible polynomial definition
  77. Field extension definition
  78. Cancellation in an integral domain statement
  79. The only ideals of a field are zero and the whole field statement
  80. Proof of the universal property of the quotient ring proof
  81. Proof that R/M is a field iff M is maximal proof
  82. Strong induction statement
  83. Function definition
  84. Principal ideal domain definition
  85. Dimension theorem for vector spaces statement
  86. Dimension definition
  87. Division algorithm for polynomials statement
  88. Degree of a product statement
  89. Algebraic element definition
  90. Preorder definition
  91. Intermediate field definition
  92. Degree of a field extension definition
  93. Universal property of the quotient ring statement
  94. Proof of the well-ordering principle from strong induction proof
  95. Proof that in a PID a nonzero element is irreducible iff it generates a maximal ideal proof
  96. Image of a set definition
  97. Preimage of a set definition
  98. Choice function definition
  99. Codomain definition
  100. Injective function definition
  101. Minimal polynomial of an algebraic element definition
  102. Partial order definition
  103. Proof that R[x] is an integral domain iff R is proof
  104. Algebraic extension definition
  105. Finite extension definition
  106. Field generated by adjoining elements definition
  107. Proof that a ring homomorphism is injective iff its kernel is zero proof
  108. Well-ordering principle statement
  109. In a PID a nonzero element is irreducible iff it generates a maximal ideal statement
  110. Axiom of choice statement
  111. Surjective function definition
  112. Maximal element definition
  113. Total order definition
  114. Upper bound definition
  115. Proof that the minimal polynomial divides every polynomial vanishing at the element proof
  116. Proof that the minimal polynomial is irreducible proof
  117. R[x] is an integral domain iff R is statement
  118. Simple extension definition
  119. A ring homomorphism is injective iff its kernel is zero statement
  120. Proof that Euclidean domains are principal ideal domains proof
  121. Bijective function definition
  122. Chain definition
  123. The minimal polynomial divides every polynomial vanishing at the element statement
  124. The minimal polynomial is irreducible statement
  125. Proof that K[x] is a Euclidean domain proof
  126. Proof of the first isomorphism theorem for rings proof
  127. Equinumerous sets definition
  128. Proof of the correspondence theorem for rings proof
  129. Euclidean domains are principal ideal domains statement
  130. Zorn's lemma statement
  131. K[x] is a Euclidean domain statement
  132. Cardinality definition
  133. First isomorphism theorem for rings statement
  134. Correspondence theorem for rings statement
  135. Proof that every linearly independent set extends to a basis, by Zorn's lemma proof
  136. Proof that every vector space has a basis, by Zorn's lemma proof
  137. R/M is a field iff M is maximal statement
  138. Proof that K[x]/(f) is a field iff f is irreducible proof
  139. Every vector space has a basis statement
  140. Every linearly independent set extends to a basis statement
  141. K[x]/(f) is a field iff f is irreducible statement
  142. Proof of the tower law proof
  143. Proof that finite extensions are algebraic proof
  144. Finite extensions are algebraic statement
  145. Tower law statement
  146. Proof that K(a) is isomorphic to K[x] modulo the minimal polynomial proof
  147. K(a) is isomorphic to K[x] modulo the minimal polynomial statement
  148. Proof that the degree of a simple extension is the degree of the minimal polynomial proof
  149. Degree of a simple extension is the degree of the minimal polynomial statement
  150. Proof that an algebraic extension of an algebraic extension is algebraic proof

Where this leads

3 pages build on this one.

Further reading

  1. Web https://www.jmilne.org/math/CourseNotes/FT.pdf, Corollary 1.31
About this page
TagX9YTFCGP9J
URIm2://X9YTFCGP9J
Kindproof
WritingW0 stub
MSC12F05
ProvesAn algebraic extension of an algebraic extension is algebraic
UsesAlgebraic extension, Algebraic element, Minimal polynomial of an algebraic element, Degree of a simple extension is the degree of the minimal polynomial, Tower law, Finite extensions are algebraic, Finite extension, Field generated by adjoining elements
Changed4 November 2026

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