Great book! The author's teaching experinece shows in every chapter. --Efim Zelmanov, University of California, San Diego Vinberg has written an algebra book that is excellent, both as a classroom text or for self-study. It is plain that years of teaching abstract algebra have enabled him to say the right thing at the right time. --Irving Kaplansky, MSRI This is a comprehensive text on modern algebra written for advanced undergraduate and basic graduate algebra classes. The book is based on courses taught by the author at the Mechanics and Mathematics Department of Moscow State University and at the Mathematical College of the Independent University of Moscow. The unique feature of the book is that it contains almost no technically difficult proofs. Following his point of view on mathematics, the author tried, whenever possible, to replace calculations and difficult deductions with conceptual proofs and to associate geometric images to algebraic objects. Another important feature is that the book presents most of the topics on several levels, allowing the student to move smoothly from initial acquaintance to thorough study and deeper understanding of the subject. Presented are basic topics in algebra such as algebraic structures, linear algebra, polynomials, groups, as well as more advanced topics like affine and projective spaces, tensor algebra, Galois theory, Lie groups, associative algebras and their representations. Some applications of linear algebra and group theory to physics are discussed. Written with extreme care and supplied with more than 200 exercises and 70 figures, the book is also an excellent text for independent study.
Author(s): E.B.Vinberg
Series: Graduate studies in Mathematics
Edition: 56
Publisher: AMS
Year: 2003
Language: English
Pages: 526
Content
Algebraic Structures
1
12 Abelian Groups
4
13 Rings and Fields
7
14 Subgroups Subrings and Subfields
10
15 The Field of Complex Numbers
12
16 Rings of Residue Classes
18
17 Vector Spaces
23
18 Algebras
27
63 Linear Operators and Bilinear Functions on Euclidean Space
212
Affine and Projective Spaces
239
72 Convex Sets
247
73 Affine Transformations and Motions
259
74 Quadrics
268
75 Projective Spaces
280
Tensor Algebra
295
82 Tensor Algebra of a Vector Space
302
19 Matrix Algebras
30
Elements of Linear Algebra
35
22 Basis and Dimension of a Vector Space
43
23 Linear Maps
53
24 Determinants
64
25 Several Applications of Determinants
76
Elements of Polynomial Algebra
81
General Properties
87
33 Fundamental Theorem of Algebra of Complex Numbers
93
34 Roots of Polynomials with Real Coefficients
98
35 Factorization in Euclidean Domains
103
36 Polynomials with Rational Coefficients
109
37 Polynomials in Several Variables
112
38 Symmetric Polynomials
116
39 Cubic Equations
123
310 Field of Rational Fractions
129
Elements of Group Theory
137
42 Groups in Geometry and Physics
143
43 Cyclic Groups
147
44 Generating Sets
153
45 Cosets
155
46 Homomorphisms
163
Vector Spaces
171
52 Linear Functions
176
53 Bilinear and Quadratic Functions
179
54 Euclidean Spaces
190
55 Hermitian Spaces
197
Linear Operators
201
62 Eigenvectors
207
83 Symmetric Algebra
308
84 Grassmann Algebra
314
Commutative Algebra
325
92 Ideals and Quotient Rings
337
93 Modules over Principal Ideal Domains
345
94 Noetherian Rings
352
95 Algebraic Extensions
356
96 Finitely Generated Algebras and Affine Algebraic Varieties
367
97 Prime Factorization
376
Groups
385
102 Commutator Subgroup
392
103 Group Actions
394
104 Sylow Theorems
400
105 Simple Groups
403
106 Galois Extensions
407
107 Fundamental Theorem of Galois Theory
412
Linear Representations and Associative Algebras
419
112 Complete Reducibility of Linear Representations of Finite and Compact Groups
430
113 FiniteDimensional Associative Algebras
434
114 Linear Representations of Finite Groups
442
115 Invariants
452
116 Division Algebras
458
Lie Groups
471
121 Definition and Simple Properties of Lie Groups
472
122 The Exponential Map
478
123 Tangent Lie Algebra and the Adjoint Representation
482
124 Linear Representations of Lie Groups
487
Answers to Selected Exercises
495
