
Algebra
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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
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