Rédei / Sneddon / Stark Algebra

1. Auflage 2014
ISBN: 978-1-4832-2264-6
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 91, 832 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Compared with the original German edition this volume contains the results of more recent research which have to some extent originated from problems raised in the previous German edition. Moreover, many minor and some important modifications have been carried out. For example paragraphs 2 - 5 were amended and their order changed. On the advice of G. Pickert, paragraph 7 has been thoroughly revised. Many improvements originate from H. J. Weinert who, by enlisting the services of a working team of the Teachers' Training College of Potsdam, has subjected large parts of this book to an exact and constructive review. This applies particularly to paragraphs 9, 50, 51, 60, 63, 66, 79, 92, 94, 97 and 100 and to the exercises. In this connection paragraphs 64 and 79 have had to be partly rewritten in consequence of the correction

Autoren/Hrsg.

Rédei, L.

Sneddon, I. N.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Algebra;4
3;Copyright Page;5
4;Table of Contents;6
5;PREFACE TO THE GERMAN EDITION;12
6;PREFACE TO THE ENGLISH EDITION;16
7;LIST OF SYMBOLS;18
8;CHAPTER I. SET-THEORETICAL PRELIMINARIES;20
8.1;§ 1. Sets;20
8.2;§ 2. Relations;22
8.3;§ 3. Mappings;23
8.4;§ 4. Multiplication of Mappings;25
8.5;§ 5. Functions;26
8.6;§ 6. Classification of a Set. Equivalence Relations;28
8.7;§ 7. Natural Numbers;30
8.8;§ 8. Equipotent Sets;33
8.9;§ 9. Ordered and Semiordered Sets;38
8.10;§ 10. Well-ordered Sets;40
8.11;§ 11*. The Lemma of Kuratowski—Zorn;41
8.12;§ 12. The Special Lemma of Kuratowski—Zorn;43
8.13;§ 13. The Lemma of Teichmiiller—Tukey;43
8.14;§ 14. The Theorem of Hausdorff—Birkhoff;44
8.15;§ 15. Theorem of Well-ordering;44
8.16;§ 16. Transfinite Induction;45
9;CHAPTER II. STRUCTURES;47
9.1;§ 17. Compositions;47
9.2;§ 18. Operators;52
9.3;§ 19. Structures;53
9.4;§ 20. Semigroups;59
9.5;§ 21. Groups;70
9.6;§ 22. Modules;76
9.7;§ 23. Rings;78
9.8;§ 24. Skew Fields;86
9.9;§ 25. Substructures;89
9.10;§ 26. Generating Elements;95
9.11;§ 27. Some Important Substructures;99
9.12;§ 28. Isomorphisms;105
9.13;§ 29. Homomorphisms;109
9.14;§ 30. Factor Structures;116
9.15;§ 31. The Homomorphy Theorem;118
9.16;§ 32. Automorphisms. Endomorphisms. Autohomomorphisms. Meromorphisms;119
9.17;§ 33. Isomorphic Structures with the Same Elements;122
9.18;§ 34. Skew Products;123
9.19;§ 35. Structure Extensions;125
9.20;§ 36. Representation of Groups by Permutation Groups;129
9.21;§ 37. Endomorphism Rings;132
9.22;§ 38. Representation of Rings by Endomorphism Rings;134
9.23;§ 39. Anti-isomorphisms. Anti-automorphisms;136
9.24;§ 40. Complexes;137
9.25;§ 41. Cosets. Residue Classes;141
9.26;§ 42. Normal Divisors. Ideals;144
9.27;§ 43. Alternating Groups;153
9.28;§ 44. Direct Products. Direct Sums;159
9.29;§ 45. Basis;171
9.30;§ 46. Congruences;173
9.31;§ 47. Quotient Structures;176
9.32;§ 48. Difference Structures;181
9.33;§ 49. Free Structures. Structures Defined by Equations;182
9.34;§ 50. Schreier Group Extensions;193
9.35;§ 51. The Holomorph of a Group;203
9.36;§ 52. Everett Ring Extensions;206
9.37;§ 53. Double Homothetisms;213
9.38;§ 54. The Holomorphs of a Ring;217
9.39;§ 55. The Two Isomorphy Theorems;219
9.40;§ 56. Simple Factor Structures;224
9.41;§ 57. Commutative Factor Structures;226
9.42;§ 58, Zassenhaus's Lemma;226
9.43;§ 59. Schreier's Main Theorem and the Jordan—Hölder Theorem;230
9.44;§ 60. Lattices;233
10;CHAPTER III. OPERATOR STRUCTURES;242
10.1;§ 61. Operator Structures;242
10.2;§ 62. Operator Groups, Operator Modules and Operator Rings;247
10.3;§ 63. Remak—Krull—Schmidt Theorem;253
10.4;§ 64. Vector Spaces. Double Vector Spaces. Algebras. Double Algebras;257
10.5;§ 65. Cross Products;270
10.6;§ 66. Monomial Rings;272
10.7;§ 67. Polynomial Rings;278
10.8;§ 68. Linear Mappings;286
10.9;§ 69. Full Matrix Rings;293
10.10;§ 70. Linear Groups;296
10.11;§ 71. Alternating Rings;299
10.12;§ 72. Determinants;301
10.13;§ 73. Cramer's Rule;309
10.14;§ 74. Characteristic Polynomials;312
10.15;§ 75. Norms and Traces;314
10.16;§ 77. The Quaternion Group;317
10.17;§ 78. Quaternion Rings;318
11;CHAPTER IV. DIVISIBILITY IN RINGS;323
11.1;§ 79. Factor Decompositions and Divisibility;323
11.2;§ 80. Ideals and Divisibility;337
11.3;§ 81. Principal Ideal Rings;340
11.4;§ 82. Euclidean Rings;344
11.5;§ 83. Euclid's Algorithm;347
11.6;§ 84. The Ring of the Integers;348
11.7;§ 85. Szendrei's Theorem;354
11.8;§ 86. Polynomial Rings over Skew Fields;356
11.9;§ 87. The Residue Theorem for Polynomials;359
11.10;§ 88. Gauss's Theorem;361
11.11;§ 89.* The Ring of Integral Quaternions;364
12;CHAPTER V. FINITE ABELIAN GROUPS;379
12.1;§ 90. Cyclic Groups;379
12.2;§ 91. Frobenius—Stickelberger Main Theorem;381
12.3;§ 92.* Hajos's Main Theorem;388
12.4;§ 93. The Character Group of Finite Abelian Groups;395
12.5;§ 94. The Mobius—Delsarte Inversion Formula;400
12.6;§ 95. Zeta Functions for Finite Abelian Groups;404
12.7;§ 96. The Group of Prime Residue Classes mod m;410
13;CHAPTER VI. OPERATOR MODULES;414
13.1;§ 97. Operator Modules and Vector Spaces;414
13.2;§ 98, Determinant Divisors and Elementary Divisors;418
13.3;§ 99. The Main Theorem for Finitely Generated Ahelian Groups;425
13.4;§ 100. Linear Dependence over Skew Fields;428
13.5;§ 101. Vector Spaces over Skew Fields;431
13.6;§ 102. Systems of Linear Equations over Skew Fields;433
13.7;§ 103. Kronecker's Rank Theorem;441
13.8;§ 104. Schur's Lemma;441
13.9;§ 105. The Density Theorem of Chevalley—Jacobson;442
13.10;§ 106. The Structure Theorems of Wedderburn—Artin;445
14;CHAPTER VII. COMMUTATIVE POLYNOMIAL RINGS;451
14.1;§ 107. McCoy's Theorem;451
14.2;§ 108. Differential Quotient;452
14.3;§ 109. Field of Rational Functions;457
14.4;§ 110. The Multiple Divisors of Polynomials;459
14.5;§ 111. Symmetric Polynomials;460
14.6;§ 112. The Resultant of Two Polynomials;462
14.7;§ 113. The Discriminant of a Polynomial;469
14.8;§ 114. The Newton Formulae;472
14.9;§ 115. Waring's Formula;473
14.10;§ 116. Interpolation;477
14.11;§ 117. Factor Decomposition According to Kronecker's Method;479
14.12;§ 118. Eisenstein's Theorem;481
14.13;§ 119, Hubert's Basis Theorem;483
14.14;§ 120,* Szekeres's Theorem;485
14.15;§ 121. Kronecker—Hensel Theorem;490
14.16;§ 122. Tschirnhaus Transformation of Ideals;492
14.17;§ 123. Rings Generated by a Single Element;494
15;CHAPTER VIII. THEORY OF FIELDS;496
15.1;§ 124. Prime Fields;496
15.2;§ 125. Relative Fields;497
15.3;§ 126. Field Extensions;500
15.4;§ 127. Simple Field Extensions;501
15.5;§ 128. Extension Fields of Finite Degree;507
15.6;§ 129. Splitting Field;509
15.7;§ 130. Steinitz's First Main Theorem;513
15.8;§ 131. Normal Fields;515
15.9;§ 132. Fields of Prime Characteristic;517
15.10;§ 133. Finite Fields;518
15.11;§ 134. König—Rados Theorem;526
15.12;§ 135. Cyclotomic Polynomials;527
15.13;§ 136. Wedderburn's Theorem;532
15.14;§ 137. Pure Transcendental Field Extensions;534
15.15;§ 138. Steinitz's Second Main Theorem;537
15.16;§ 139. Simple Transcendental Field Extension;542
15.17;§ 140. Isomorphisms of an Algebraic Field;546
15.18;§ 141. Separable and Inseparable Field Extensions;551
15.19;§ 142. Complete and Incomplete Fields;560
15.20;§ 143. Simplicity of Field Extensions;567
15.21;§ 144. Norms and Traces in Fields of Finite Degree;570
15.22;§ 145. Differents and Discriminants in Separable Fields of Finite Degree;574
15.23;§ 146. Ore Polynomial Rings;577
15.24;§ 147.* Normal Bases of Finite Fields;579
16;CHAPTER IX. ORDERED STRUCTURES;587
16.1;§ 148. Ordered Structures;587
16.2;§ 149. Archimedean and Non-Archimedean Orderings;599
16.3;§ 150. Absolute Value in Ordered Structures;602
17;CHAPTER X. FIELDS WITH VALUATION;604
17.1;§ 151. Valuations;604
17.2;§ 152. Convergent Sequences and Limits;606
17.3;§ 153. Perfect Hull;613
17.4;§ 154. The Field of Real Numbers;621
17.5;§ 155. The Field of Complex Numbers;629
17.6;§ 156. Really Closed Fields;634
17.7;§ 157. Archimedean and Non-Archimedean Valuations;636
17.8;§ 158. Exponent Valuations;638
17.9;§ 159. Discrete Valuations;646
17.10;§ 160. p-adic Valuations;648
17.11;§ 161. Ostrowski's First Theorem;652
17.12;§ 162. HensePs Lemma;654
17.13;§ 163. Extensions of Real Perfect Valuations for Field Extensions of Finite Degree;658
17.14;§ 164. Ostrowski's Second Theorem;663
17.15;§ 165. Extensions of Real Valuations for Algebraic Field Extensions;666
17.16;§ 166. Real Valuations of Number Fieldsof Finite Degree;667
17.17;§ 167. Real Valuations of Simple TranscendentalField Extensions;668
18;CHAPTER XI. GALOIS THEORY;674
18.1;§ 168. Fundamental Theorem of Galois Theory;674
18.2;§ 169. Stickelberger's Theorem on Finite Fields;682
18.3;§ 170. The Quadratic Reciprocity Theorem;683
18.4;§ 171. Cyclotomic Fields;688
18.5;§ 172. Cyclic Fields;691
18.6;§ 173. Solvable Equations;699
18.7;§ 174. The General Algebraic Equation;706
18.8;§ 175. Tschirnhaus Transformation of Polynomials;710
18.9;§ 176. Equations of Second, Third and Fourth Degree;711
18.10;§ 177. The Irreducible Case;720
18.11;§ 178. Equations of Third and Fourth Degree over Finite Fields;722
18.12;§ 179. Geometrical Constructibility;727
18.13;§ 180*. Remarkable Points of the Triangle;732
18.14;§ 181. Determination of the Galois Group of an Equation;748
18.15;§ 182. Normal Bases;752
19;CHAPTER XII. FINITE ONE-STEPNON-COMMUTATIVE STRUCTURES;755
19.1;§ 183.* Finite One-step Non-commutative Groups;755
19.2;§ 184.* Finite One-step Non-commutative Rings;772
19.3;§ 185.* Finite One-step Non-commutative Semigroups;805
20;BIBLIOGRAPHY;818
21;INDEX;828
22;OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS;840