Basic Notions of Algebra, Volume 1, Issue 1

Front Cover
Igor R. Shafarevich
Springer Science & Business Media, Apr 13, 2005 - Mathematics - 258 pages
1 Review
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.
  

What people are saying - Write a review

User Review - Flag as inappropriate

下载地址: http://www.eknigu.com/info/M_Mathematics/MA_Algebra/MAa_Abstract%20algebra/Kostrikin%20A.I.,%20Shafarevich%20I.R.%20Algebra%20I.%20Basic%20notions%20of%20algebra%20(Enc.Math.Sci.11,%20Springer,%202005)(T)(260s).djvu

Contents

1 What is Algebra?
6
2 Fields
11
3 Commutative Rings
17
4 Homomorphisms and Ideals
24
5 Modules
33
6 Algebraic Aspects of Dimension
41
7 The Algebraic View of Infinitesimal Notions
50
8 Noncommutative Rings
61
C Representations of the Classical Complex Lie Groups
176
18 Some Applications of Groups
178
B The Galois Theory of Linear Differential Equations PicardVessiot Theory
182
C Classification of Unramified Covers
183
D Invariant Theory
184
E Group Representations and the Classification of Elementary Particles
186
19 Lie Algebras and Nonassociative Algebra
189
B Lie Theory
193

9 Modules over Noncommutative Rings
74
10 Semisimple Modules and Rings
79
11 Division Algebras of Finite Rank
90
12 The Notion of a Group
96
Finite Groups
108
Infinite Discrete Groups
125
Lie Groups and Algebraic Groups
141
A Compact Lie Groups
144
B Complex Analytic Lie Groups
148
C Algebraic Groups
151
16 General Results of Group Theory
152
17 Group Representations
161
A Representations of Finite Groups
164
B Representations of Compact Lie Groups
168
C Applications of Lie Algebras
198
D Other Nonassociative Algebras
200
20 Categories
203
21 Homological Algebra
214
B Cohomology of Modules and Groups
220
C Sheaf Cohomology
226
22 Ktheory
231
B Algebraic Ktheory
235
Comments on the Literature
240
References
245
Index of Names
250
Subject Index
252
Copyright

Common terms and phrases

References to this book

All Book Search results »

Bibliographic information