Proof that the algebraic numbers form a field

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 the algebraic numbers form a field proof