Algebra I: Basic Notions of Algebra (Encyclopaedia of Mathematical Sciences) (English and Russian Edition)

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. 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 From the reviews: "This is one of the few mathematical books, the reviewer has read from cover to cover ... The main merit is that nearly on every page you will find some unexpected insights..." Zentralblatt für Mathematik und Ihre Grenzgebiete, 1991 "...which I read like a novel and undoubtedly will become a classic. ... A merit of the book under review is that it contains several important articles from journals which are not all so easily accessible. ... Furthermore, at the end of the book, there are some Notes by the author which are indispensible for the necessary historical background information. ... This valuable book should be on the shelf of every algebraist and algebraic geometer." Nieuw Archief voor Wiskunde, 1992 "... There are few proofs in full, but there is an exhilarating combination of sureness of foot and lightness of touch in the exposition ... which transports the reader effortlessly across the whole spectrum of algebra.... The challenge to Ezekiel, "Can these bones live?" is, all too often, the reaction of students when introduced to the bare bones of the concepts and constructs of modern algebra. Shafarevich's book - which reads as comfortably as an extended essay - breathes life into the skeleton and will be of interest to many classes of readers..." The Mathematical Gazette, 1991 "... According to the preface, the book is addressed to "students of mathematics in the first years of an undergraduate course, or theoretical physicists or mathematicians from outside algebra wanting to get an impression of the spirit of algebra and its place in mathematics." I think that this promise is fully justified. The beginner, the experts and also the interested scientist who had contact with algebraic notions - all will read this exceptional book with great pleasure and benefit." Zeitschrift für Kristallographie, 1991 §22. K-theory 230 A. Topological X-theory 230 Vector bundles and the functor Vec(X). Periodicity and the functors KJX). K(X) and t the infinite-dimensional linear group. The symbol of an elliptic differential operator. The index theorem. B. Algebraic K-theory 234 The group of classes of projective modules. K, K and K of a ring. K of a field and o l n 2 its relations with the Brauer group. K-theory and arithmetic. Comments on the Literature 239 References 244 Index of Names 249 Subject Index 251 Preface This book aims to present a general survey of algebra, of its basic notions and main branches. Now what language should we choose for this? In reply to the question'What does mathematics study?', it is hardly acceptable to answer'structures'or'sets with specified relations'; for among the myriad conceivable structures or sets with specified relations, only a very small discrete subset is of real interest to mathematicians, and the whole point of the question is to understand the special value of this infinitesimal fraction dotted among the amorphous masses. In the same way, the meaning of a mathematical notion is by no means confined to its formal definition; in fact, it may be rather better expressed by a (generally fairly small) sample of the basic examples, which serve the mathematician as the motivation and the substantive definition, and at the same time as the real meaning of the notion. Wholeheartedly recommended to every student and user of mathematics, this is an extremely original and highly informative essay on algebra and its place in modern mathematics and science. From the fields studied in every university maths course, through Lie groups to cohomology and category theory, the author shows how the origins of each concept can be related to attempts to model phenomena in physics or in other branches of mathematics. Required reading for mathematicians, from beginners to experts. From the fields, commutative rings and groups studied in university mathematics courses, through Lie groups and algebras to category theory, this text shows how the origins of each algebraic concept can be related to attempts to model phenomena in physics or in other branches of mathematics. A.i. Kostrikin, I.r. Shafarevich, (eds.). Translation Of: Algebra 1, Issued As Part Of The Series: Itogi Nauki I Tekhniki. Serii︠a︡ Sovremennye Problemy Matematiki. Fundamentalʹnye Napravlenii︠a︡. Includes Bibliographical References.