Algebra, Vol.1

Title Page......Page 3
Copyright page......Page 4
Contents......Page 5
Preface to the German edition......Page 11
Preface to the English edition......Page 15
List of Symbols......Page 17
1 .Sets......Page 19
2. Relations......Page 21
3. Mappings......Page 22
4. Multiplication of mappings......Page 24
5. Functions......Page 25
6. Classification of a set. Equivalence relations......Page 27
7. Natural numbers......Page 29
8. Equipotent sets......Page 32
9. Ordered and semiordered sets......Page 37
10. Well-ordered sets......Page 39
11. The lemma of Kuratowski- Zorn......Page 40
13. The lemma of Teichmuller-Tukey......Page 42
15. Theorem of well-ordering......Page 43
16. Transfinite induction......Page 44
17. Compositions......Page 46
18. Operators......Page 51
19. Structures......Page 52
20. Semigroups......Page 58
21. Groups......Page 69
22. Modules......Page 75
23. Rings......Page 77
24. Skew fields......Page 85
25. Substructures......Page 88
26. Generating elements......Page 94
27. Some important substructures......Page 98
28. [somorphisms......Page 104
29. Homomorphisms......Page 108
30. Factor structures......Page 115
31. The homomorphy theorem......Page 117
32. Automorphisms. Endomorphisms. Autohomomorphisms. Meromorphisms......Page 118
33. Isomorphic structures with the same elements......Page 121
34. Skew products......Page 122
35. Structure extensions......Page 124
36. Representation of groups by permutation groups......Page 128
37. Endomorphism rings......Page 131
38. Representation of rings by endomorphism rings......Page 133
39. Anti-isomorphisms. Anti-automorphisms......Page 135
40. Complexes......Page 136
41. Cosets. Residue classes......Page 140
42. Normal divisors. Ideals......Page 143
43. Alternating groups......Page 152
44. Direct products. Direct sums......Page 158
45. Basis......Page 170
46. Congruences......Page 172
47. Quotient structures......Page 175
48. Difference structures......Page 180
49. Free structures. Structures defined by equations......Page 181
50. Schreier group extensions......Page 192
51. The holomorph of a group......Page 202
52. Everett ring extensions......Page 205
53. Double homothetisms......Page 212
54. The holomorphs of a ring......Page 216
55. The two isomorphy theorems......Page 218
56. Simple factor structures......Page 223
58. Zassenhaus's lemma......Page 225
59. Schreier's main theorem and the Jordan-Holder theorem......Page 229
60. Lattices......Page 232
61. Operator structures......Page 241
62. Operator groups, operator modules and operator rings......Page 246
63. Remak-Krull-Schmidt theorem......Page 252
64. Vector spaces. Double vector spaces. Algebras. Double algebras......Page 256
65. Cross products......Page 269
66. Monomial rings......Page 271
67. Polynomial rings......Page 277
68. Linear mappings......Page 285
69. Full matrix rings......Page 292
70. Linear groups......Page 295
71. Alternating rings......Page 298
72. Determinants......Page 300
73. Cramer's rule......Page 308
74. Characteristic polynomials......Page 311
75. Norms and traces......Page 313
76. Complex rings......Page 315
77. The quaternion group......Page 316
78. Quaternion rings......Page 317
79. Factor decompositions and divisibility......Page 322
80. Ideals and divisibility......Page 336
81. Principal ideal rings......Page 339
82. Euclidean rings......Page 343
83. Euclid's algorithm......Page 346
84. The ring of the integers......Page 347
85. Szendrei's theorem......Page 353
86. Polynomial rings over skew fields......Page 355
87. The residue theorem for polynomials......Page 358
88. Gauss's theorem......Page 360
89. The ring of integral quaternions......Page 363
90. Cyclic groups......Page 378
91. Frobenius-Stickelberger main theorem......Page 380
92. Haj6s's main theorem......Page 387
93. The character group of finite Abelian groups......Page 394
94. The Mdbius-Delsarte inversion formula......Page 399
95. Zeta functions for finite Abelian groups......Page 403
96. The group of prime residue classes mod in......Page 409
97. Operator modules and vector spaces......Page 413
98. Determinant divisors and elementary divisors......Page 417
99. The main theorem for finitely generated Abelian groups......Page 424
100. Linear dependence over skew fields......Page 427
101. Vector spaces over skew fields            ......Page 430
102. Systems of linear equations over skew fields......Page 432
104. Schur's lemma......Page 440
105. The density theorem of Chevalley-Jacobson......Page 441
106. The structure theorems of Wedderburn-Artin......Page 444
107. McCoy's theorem......Page 350
108. Differential quotient......Page 451
109. Field of rational functions......Page 456
110. The multiple divisors of polynomials......Page 458
111. Symmetric polynomials......Page 459
112. The resultant of two polynomials......Page 461
113. The discriminant of a polynomial......Page 468
114. The Newton formulae......Page 471
115. Waring's formula......Page 472
116. Interpolation......Page 476
117. Factor decomposition according to Kronecker's method......Page 478
118. Eisenstein's theorem......Page 480
119. Hilbert's basis theorem......Page 482
120. Szekeres's theorem......Page 484
121. Kronecker-Hensel theorem......Page 489
122. Tschirnhaus transformation of ideals......Page 491
123. Rings generated by a single element......Page 493
124. Prime fields......Page 495
125. Relative fields......Page 496
126. Field extensions......Page 499
127. Simple field extensions......Page 500
128. Extension fields of finite degree......Page 506
129. Splitting field......Page 508
130. Steinitz's first main theorem......Page 512
131. Normal fields......Page 514
132. Fields of prime characteristic......Page 516
133. Finite fields......Page 517
134. Kong-Rados theorem......Page 525
135. Cyclotomic polynomials......Page 526
136. Wedderburn's theorem......Page 531
137. Pure transcendental field extensions......Page 533
138. Steinitz's second main theorem......Page 536
139. Simple transcendental field extensions......Page 541
140. Isomorphisms of an algebraic field......Page 545
141. Separable and inseparable field extensions......Page 550
142. Complete and incomplete fields......Page 559
143. Simplicity of field extensions......Page 566
144. Norms and traces in fields of finite degree......Page 569
145. Differents and discriminants in separable fields of finite degree......Page 573
146. Ore polynomial rings......Page 576
147. Normal bases of finite fields......Page 578
148. Ordered structures......Page 586
149. Archimedean and non-Archimedean orderings......Page 598
150. Absolute value in ordered structures......Page 601
151. Valuations......Page 603
152. Convergent sequences and limits......Page 605
153. Perfect hull......Page 612
154. The field of real numbers......Page 620
155. The field of complex numbers......Page 628
156. Really closed fields......Page 633
157. Archimedean and non-Archimedean valuations......Page 635
158. Exponent valuations......Page 637
159. Discrete valuations......Page 645
160. p-adic valuations......Page 647
161. Ostrowski's first theorem......Page 651
162. Hensel's lemma......Page 653
163. Extensions of real perfect valuations for field extensions of finite degree......Page 657
164. Ostrowski's second theorem......Page 662
165. Extensions of real valuations for algebraic field extensions......Page 665
166. Real valuations of number fields of finite degree......Page 666
167. Real valuations of simple transcendental field extensions......Page 667
168. Fundamental theorem of Galois theory......Page 673
169. Stickelberger's theorem on finite fields......Page 681
170. The quadratic reciprocity theorem......Page 682
171. Cyclotomic fields......Page 687
172. Cyclic fields......Page 690
173. Solvable equations......Page 698
174. The general algebraic equation......Page 705
175. Tschirnhaus transformation of polynomials......Page 709
176. Equations of second, third and fourth degree......Page 710
177. The irreducible case......Page 719
178. Equations of third and fourth degree over finite fields......Page 721
179. Geometrical constructibility......Page 726
180. Remarkable points of the triangle......Page 731
181. Determination of the Galois group of an equation......Page 747
182. Normal bases......Page 751
183. Finite one-step non-commutative groups......Page 754
184. Finite one-step non-commutative rings......Page 771
185. Finite one-step non-commutative semigroups......Page 804
Bibliography......Page 817
Index......Page 827
Other titles in the series......Page 839