This book is wholeheartedly recommended to every student or user of mathematics. Although the author modestly describes his book as 'merely an attempt to talk about' algebra, he succeeds in writing an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields, commutative rings and groups studied in every university math course, through Lie groups and algebras to cohomology and category theory, the author shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Comparable in style with Hermann Weyl's evergreen essay The Classical Groups, Shafarevich's new book is sure to become required reading for mathematicians, from beginners to experts.
Author(s): Igor R. Shafarevich, Aleksej I. Kostrikin, M. Reid
Year: 1990
Language: English
Pages: 262
Contents......Page 5
Preface......Page 8
§ 1. What is Algebra?......Page 10
§ 2. Fields......Page 15
§ 3. Commutative Rings......Page 21
§ 4. Homomorphisms and Ideals......Page 28
§ 5. Modules......Page 37
§ 6. Algebraic Aspects of Dimension......Page 45
§ 7. The Algebraic View of Infinitesimal Notions......Page 54
§ 8. Noncommutative Rings......Page 65
§ 9. Modules over Noncommutative Rings......Page 78
§ 10. Semisimple Modules and Rings......Page 83
§ 11. Division Algebras of Finite Rank......Page 94
§ 12. The Notion of a Group......Page 100
§ 13. Examples of Groups: Finite Groups......Page 112
§ 14. Examples of Groups: Infinite Discrete Groups......Page 128
§ 15. Examples of Groups: Lie Groups and Algebraic Groups......Page 144
A. Compact Lie Groups......Page 147
B. Complex Analytic Lie Groups......Page 151
C. Algebraic Groups......Page 154
§ 16. General Results of Group Theory......Page 155
§ 17. Group Representations......Page 164
A. Representations of Finite Groups......Page 167
B. Representations of Compact Lie Groups......Page 171
C. Representations of the Classical Complex Lie Groups......Page 179
A. Galois Theory......Page 181
B. The Galois Theory of Linear Differential Equations (Picard-Vessiot Theory)......Page 185
C. Classification of Unramified Covers......Page 186
D. Invariant Theory......Page 187
E. Group Representations and the Classification of Elementary Particles......Page 189
A. Lie Algebras......Page 192
B. Lie Theory......Page 196
C. Applications of Lie Algebras......Page 201
D. Other Nonassociative Algebras......Page 203
§ 20. Categories......Page 206
A. Topological Origins of the Notions of Homological Algebra......Page 217
B. Cohomology of Modules and Groups......Page 223
C. Sheaf Cohomology......Page 229
A. Topological K-theory......Page 234
B. Algebraic K-theory......Page 238
Comments on the Literature......Page 243
References......Page 248
K......Page 253
Z......Page 254
C......Page 255
D......Page 256
F......Page 257
I......Page 258
M......Page 259
Q......Page 260
S......Page 261
Z......Page 262
