Algebra
Published on 21. July 2014
832 pages
978-1-4832-2264-6 (ISBN)
Description
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
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Content
- Front Cover
- Algebra
- Copyright Page
- Table of Contents
- PREFACE TO THE GERMAN EDITION
- PREFACE TO THE ENGLISH EDITION
- LIST OF SYMBOLS
- CHAPTER I. SET-THEORETICAL PRELIMINARIES
- § 1. Sets
- § 2. Relations
- § 3. Mappings
- § 4. Multiplication of Mappings
- § 5. Functions
- § 6. Classification of a Set. Equivalence Relations
- § 7. Natural Numbers
- § 8. Equipotent Sets
- § 9. Ordered and Semiordered Sets
- § 10. Well-ordered Sets
- § 11*. The Lemma of Kuratowski-Zorn
- § 12. The Special Lemma of Kuratowski-Zorn
- § 13. The Lemma of Teichmiiller-Tukey
- § 14. The Theorem of Hausdorff-Birkhoff
- § 15. Theorem of Well-ordering
- § 16. Transfinite Induction
- CHAPTER II. STRUCTURES
- § 17. Compositions
- § 18. Operators
- § 19. Structures
- § 20. Semigroups
- § 21. Groups
- § 22. Modules
- § 23. Rings
- § 24. Skew Fields
- § 25. Substructures
- § 26. Generating Elements
- § 27. Some Important Substructures
- § 28. Isomorphisms
- § 29. Homomorphisms
- § 30. Factor Structures
- § 31. The Homomorphy Theorem
- § 32. Automorphisms. Endomorphisms. Autohomomorphisms. Meromorphisms
- § 33. Isomorphic Structures with the Same Elements
- § 34. Skew Products
- § 35. Structure Extensions
- § 36. Representation of Groups by Permutation Groups
- § 37. Endomorphism Rings
- § 38. Representation of Rings by Endomorphism Rings
- § 39. Anti-isomorphisms. Anti-automorphisms
- § 40. Complexes
- § 41. Cosets. Residue Classes
- § 42. Normal Divisors. Ideals
- § 43. Alternating Groups
- § 44. Direct Products. Direct Sums
- § 45. Basis
- § 46. Congruences
- § 47. Quotient Structures
- § 48. Difference Structures
- § 49. Free Structures. Structures Defined by Equations
- § 50. Schreier Group Extensions
- § 51. The Holomorph of a Group
- § 52. Everett Ring Extensions
- § 53. Double Homothetisms
- § 54. The Holomorphs of a Ring
- § 55. The Two Isomorphy Theorems
- § 56. Simple Factor Structures
- § 57. Commutative Factor Structures
- § 58, Zassenhaus's Lemma
- § 59. Schreier's Main Theorem and the Jordan-Hölder Theorem
- § 60. Lattices
- CHAPTER III. OPERATOR STRUCTURES
- § 61. Operator Structures
- § 62. Operator Groups, Operator Modules and Operator Rings
- § 63. Remak-Krull-Schmidt Theorem
- § 64. Vector Spaces. Double Vector Spaces. Algebras. Double Algebras
- § 65. Cross Products
- § 66. Monomial Rings
- § 67. Polynomial Rings
- § 68. Linear Mappings
- § 69. Full Matrix Rings
- § 70. Linear Groups
- § 71. Alternating Rings
- § 72. Determinants
- § 73. Cramer's Rule
- § 74. Characteristic Polynomials
- § 75. Norms and Traces
- § 77. The Quaternion Group
- § 78. Quaternion Rings
- CHAPTER IV. DIVISIBILITY IN RINGS
- § 79. Factor Decompositions and Divisibility
- § 80. Ideals and Divisibility
- § 81. Principal Ideal Rings
- § 82. Euclidean Rings
- § 83. Euclid's Algorithm
- § 84. The Ring of the Integers
- § 85. Szendrei's Theorem
- § 86. Polynomial Rings over Skew Fields
- § 87. The Residue Theorem for Polynomials
- § 88. Gauss's Theorem
- § 89.* The Ring of Integral Quaternions
- CHAPTER V. FINITE ABELIAN GROUPS
- § 90. Cyclic Groups
- § 91. Frobenius-Stickelberger Main Theorem
- § 92.* Hajos's Main Theorem
- § 93. The Character Group of Finite Abelian Groups
- § 94. The Mobius-Delsarte Inversion Formula
- § 95. Zeta Functions for Finite Abelian Groups
- § 96. The Group of Prime Residue Classes mod m
- CHAPTER VI. OPERATOR MODULES
- § 97. Operator Modules and Vector Spaces
- § 98, Determinant Divisors and Elementary Divisors
- § 99. The Main Theorem for Finitely Generated Ahelian Groups
- § 100. Linear Dependence over Skew Fields
- § 101. Vector Spaces over Skew Fields
- § 102. Systems of Linear Equations over Skew Fields
- § 103. Kronecker's Rank Theorem
- § 104. Schur's Lemma
- § 105. The Density Theorem of Chevalley-Jacobson
- § 106. The Structure Theorems of Wedderburn-Artin
- CHAPTER VII. COMMUTATIVE POLYNOMIAL RINGS
- § 107. McCoy's Theorem
- § 108. Differential Quotient
- § 109. Field of Rational Functions
- § 110. The Multiple Divisors of Polynomials
- § 111. Symmetric Polynomials
- § 112. The Resultant of Two Polynomials
- § 113. The Discriminant of a Polynomial
- § 114. The Newton Formulae
- § 115. Waring's Formula
- § 116. Interpolation
- § 117. Factor Decomposition According to Kronecker's Method
- § 118. Eisenstein's Theorem
- § 119, Hubert's Basis Theorem
- § 120,* Szekeres's Theorem
- § 121. Kronecker-Hensel Theorem
- § 122. Tschirnhaus Transformation of Ideals
- § 123. Rings Generated by a Single Element
- CHAPTER VIII. THEORY OF FIELDS
- § 124. Prime Fields
- § 125. Relative Fields
- § 126. Field Extensions
- § 127. Simple Field Extensions
- § 128. Extension Fields of Finite Degree
- § 129. Splitting Field
- § 130. Steinitz's First Main Theorem
- § 131. Normal Fields
- § 132. Fields of Prime Characteristic
- § 133. Finite Fields
- § 134. König-Rados Theorem
- § 135. Cyclotomic Polynomials
- § 136. Wedderburn's Theorem
- § 137. Pure Transcendental Field Extensions
- § 138. Steinitz's Second Main Theorem
- § 139. Simple Transcendental Field Extension
- § 140. Isomorphisms of an Algebraic Field
- § 141. Separable and Inseparable Field Extensions
- § 142. Complete and Incomplete Fields
- § 143. Simplicity of Field Extensions
- § 144. Norms and Traces in Fields of Finite Degree
- § 145. Differents and Discriminants in Separable Fields of Finite Degree
- § 146. Ore Polynomial Rings
- § 147.* Normal Bases of Finite Fields
- CHAPTER IX. ORDERED STRUCTURES
- § 148. Ordered Structures
- § 149. Archimedean and Non-Archimedean Orderings
- § 150. Absolute Value in Ordered Structures
- CHAPTER X. FIELDS WITH VALUATION
- § 151. Valuations
- § 152. Convergent Sequences and Limits
- § 153. Perfect Hull
- § 154. The Field of Real Numbers
- § 155. The Field of Complex Numbers
- § 156. Really Closed Fields
- § 157. Archimedean and Non-Archimedean Valuations
- § 158. Exponent Valuations
- § 159. Discrete Valuations
- § 160. p-adic Valuations
- § 161. Ostrowski's First Theorem
- § 162. HensePs Lemma
- § 163. Extensions of Real Perfect Valuations for Field Extensions of Finite Degree
- § 164. Ostrowski's Second Theorem
- § 165. Extensions of Real Valuations for Algebraic Field Extensions
- § 166. Real Valuations of Number Fieldsof Finite Degree
- § 167. Real Valuations of Simple TranscendentalField Extensions
- CHAPTER XI. GALOIS THEORY
- § 168. Fundamental Theorem of Galois Theory
- § 169. Stickelberger's Theorem on Finite Fields
- § 170. The Quadratic Reciprocity Theorem
- § 171. Cyclotomic Fields
- § 172. Cyclic Fields
- § 173. Solvable Equations
- § 174. The General Algebraic Equation
- § 175. Tschirnhaus Transformation of Polynomials
- § 176. Equations of Second, Third and Fourth Degree
- § 177. The Irreducible Case
- § 178. Equations of Third and Fourth Degree over Finite Fields
- § 179. Geometrical Constructibility
- § 180*. Remarkable Points of the Triangle
- § 181. Determination of the Galois Group of an Equation
- § 182. Normal Bases
- CHAPTER XII. FINITE ONE-STEPNON-COMMUTATIVE STRUCTURES
- § 183.* Finite One-step Non-commutative Groups
- § 184.* Finite One-step Non-commutative Rings
- § 185.* Finite One-step Non-commutative Semigroups
- BIBLIOGRAPHY
- INDEX
- OTHER TITLES IN THE SERIES IN PURE AND APPLIED MATHEMATICS
