Proof that the multiplicative group of a finite field is cyclic

Reading path

To understand this page from the ground up, read these 139 pages in order. The foundations come first. The path takes the proofs Proof of Bézout's identity, Proof of cancellation in an integral domain, Proof of the generalized Euclid's lemma, Proof of the commutativity of addition by induction, Proof that equivalence classes form a partition, Proof that g to the n equals 1 iff the order divides n, Proof that R[x] is an integral domain iff R is, Proof that g to the order of the group is the identity, Proof that finite subgroups of the multiplicative group of a field are cyclic, Proof that the order of an element divides the order of the group, Proof of the degree of a product over an integral domain, Proof of strong induction from ordinary induction, Proof of the laws of exponents in a group, Proof that a product of two finite cyclic groups is cyclic iff their orders are coprime, Proof that a polynomial over a field has at most as many roots as its degree, Proof of the remainder theorem, Proof that a finite abelian group is cyclic iff its exponent equals its order, Proof that the order of a cyclic subgroup equals the order of its generator, Proof of the well-ordering principle from strong induction, Proof of the division algorithm, Proof of the uniqueness of inverses, Proof that cyclic groups of the same order are isomorphic, Proof of Lagrange's theorem, Proof of the factor theorem, Proof that cosets have the same cardinality as the subgroup.

1. Binary operation definition
2. Definition by recursion concept
3. Divisibility definition
4. Element definition
5. Group concept
6. Left coset concept
7. Mathematical induction concept
8. Natural numbers object
9. Peano axioms concept
10. Proposition concept
11. Vector space definition
12. Commutativity definition
13. Identity element definition
14. Greatest common divisor definition
15. Set definition
16. Direct product of groups definition
17. Group homomorphism definition
18. Order of a group definition
19. Ring definition
20. Subgroup definition
21. Successor function definition
22. Conjunction definition
23. Disjunction definition
24. Implication definition
25. Predicate definition
26. External direct sum definition
27. Inverse element definition
28. Abelian group definition
29. Coprime integers definition
30. Empty set object
31. Family of sets definition
32. Ordered pair definition
33. Group isomorphism definition
34. Intersection definition
35. Addition of natural numbers definition
36. Union definition
37. Commutative ring definition
38. Polynomial ring definition
39. Ring homomorphism definition
40. Unit of a ring definition
41. Zero divisor definition
42. Left coset definition
43. Right coset definition
44. Set-builder notation concept
45. The successor function is injective statement
46. Zero is not a successor statement
47. Biconditional definition
48. Existential quantifier definition
49. Universal quantifier definition
50. Proof of the uniqueness of inverses proof
51. Cartesian product definition
52. Disjoint sets definition
53. Order on the natural numbers definition
54. Proof of the commutativity of addition by induction proof
55. Group of units of a ring definition
56. Divisibility in a commutative ring definition
57. Subset definition
58. Degree of a polynomial definition
59. Polynomial evaluation definition
60. Union of a family of sets definition
61. Field definition
62. Axiom of extensionality statement
63. Integral domain definition
64. Uniqueness quantifier definition
65. Uniqueness of inverses statement
66. Binary relation definition
67. Commutativity of addition statement
68. Partition of a set definition
69. Leading coefficient definition
70. Root of a polynomial definition
71. Proof of cancellation in an integral domain proof
72. Evaluation homomorphism definition
73. Proof of strong induction from ordinary induction proof
74. Finite field definition
75. Reflexive relation definition
76. Symmetric relation definition
77. Transitive relation definition
78. Proof of the laws of exponents in a group proof
79. Monic polynomial definition
80. Strong induction statement
81. Multiplicity of a root definition
82. Function definition
83. Proof of the degree of a product over an integral domain proof
84. Proof of the remainder theorem proof
85. Cancellation in an integral domain statement
86. Equivalence relation definition
87. Laws of exponents in a group statement
88. Proof of the well-ordering principle from strong induction proof
89. Codomain definition
90. Injective function definition
91. Division algorithm for polynomials statement
92. Degree of a product statement
93. Order of an element definition
94. Well-ordering principle statement
95. Cyclic subgroup generated by an element definition
96. Surjective function definition
97. Remainder theorem statement
98. Equivalence class definition
99. Proof that R[x] is an integral domain iff R is proof
100. Exponent of a group definition
101. Proof of the division algorithm proof
102. Cyclic group definition
103. Bijective function definition
104. Proof that equivalence classes form a partition proof
105. Proof of the factor theorem proof
106. R[x] is an integral domain iff R is statement
107. Division algorithm statement
108. Equinumerous sets definition
109. Equivalence classes form a partition statement
110. Factor theorem statement
111. Cardinality definition
112. Proof of Bézout's identity proof
113. Proof that g to the n equals 1 iff the order divides n proof
114. Proof that a polynomial over a field has at most as many roots as its degree proof
115. g to the n equals 1 iff the order divides n statement
116. Bézout's identity statement
117. A polynomial over a field has at most as many roots as its degree statement
118. Proof that cosets have the same cardinality as the subgroup proof
119. Proof of the generalized Euclid's lemma proof
120. Proof that the order of a cyclic subgroup equals the order of its generator proof
121. Cosets have the same cardinality as the subgroup statement
122. Generalized Euclid's lemma statement
123. Order of a cyclic subgroup equals the order of its generator statement
124. Proof of Lagrange's theorem proof
125. Proof that a product of two finite cyclic groups is cyclic iff their orders are coprime proof
126. Proof that cyclic groups of the same order are isomorphic proof
127. Lagrange's theorem statement
128. Proof that the order of an element divides the order of the group proof
129. Cyclic groups of the same order are isomorphic statement
130. Product of cyclic groups is cyclic iff the orders are coprime statement
131. Order of an element divides the order of the group statement
132. Proof that g to the order of the group is the identity proof
133. Classification of finite abelian groups statement
134. g to the order of the group is the identity statement
135. Proof that a finite abelian group is cyclic iff its exponent equals its order proof
136. A finite abelian group is cyclic iff its exponent equals its order statement
137. Proof that finite subgroups of the multiplicative group of a field are cyclic proof
138. Finite subgroups of the multiplicative group of a field are cyclic statement
139. Proof that the multiplicative group of a finite field is cyclic proof