A Course in Algebra (Graduate Studies in Mathematics, Vol. 56) (Graduate Studies in Mathematics, V. 56)

This is a comprehensive textbook on modern algebra written by an internationally renowned specialist. It covers material traditionally found in advanced undergraduate and basic graduate courses and presents it in a lucid style. The author includes almost no technically difficult proofs, and reflecting his point of view on mathematics, he tries wherever possible to replace calculations and difficult deductions with conceptual proofs and to associate geometric images to algebraic objects. The effort spent on the part of students in absorbing these ideas will pay off when they turn to solving problems outside of this textbook. Another important feature is the presentation of most topics on several levels, allowing students to move smoothly from initial acquaintance with the subject to thorough study and a deeper understanding. Basic topics are included, such as algebraic structures, linear algebra, polynomials, and groups, as well as more advanced topics, such as affine and projective spaces, tensor algebra, Galois theory, Lie groups, and associative algebras and their representations. Some applications of linear algebra and group theory to physics are discussed. The book is written with extreme care and contains over 200 exercises and 70 figures. It is ideal as a textbook and also suitable for independent study for advanced undergraduates and graduate students. American Mathematical Society Front Cover 1 Title Page 4 Copyright Page 5 Contents 6 Preface 10 Chapter 1. Algebraic Structures 12 1.1. Introduction 12 1.2. Abelian Groups 15 1.3. Rings and Fields 18 1.4. Subgroups, Subrings, and Subfields 21 1.5. The Field of Complex Numbers 23 1.6. Rings of Residue Classes 29 1.7. Vector Spaces 34 1.8. Algebras 38 1.9. Matrix Algebra 41 Chapter 2. Elements of Linear Algebra 46 2.1. Systems of Linear Equations 46 2.2. Basis and Dimension of a Vector Space 54 2.3. Linear Maps 64 2.4. Determinants 75 2.5. Several Applications of Determinants 87 Chapter 3. Elements of Polynomial Algebra 92 3.1. Polynomial Algebra: Construction and Basic Properties 92 3.2. Roots of Polynomials: General Properties 98 3.3. Fundamental Theorem of Algebra of Complex Numbers 104 3.4. Roots of Polynomials with Real Coefficients 109 3.5. Factorization in Euclidean Domains 114 3.6. Polynomials with Rational Coefficients 120 3.7. Polynomials in Several Variables 123 3.8. Symmetric Polynomials 127 3.9. Cubic Equations 134 3.10. Field of Rational Fractions 140 Chapter 4. Elements of Group Theory 148 4.1. Definitions and Examples 148 4.2. Groups in Geometry and Physics 154 4.3. Cyclic Groups 158 4.4. Generating Sets 164 4.5. Cosets 166 4.6. Homomorphisms 174 Chapter 5. Vector Spaces 182 5.1. Relative Position of Subspaces 182 5.2. Linear Functions 187 5.3. Bilinear and Quadratic Functions 190 5.4. Euclidean Spaces 201 5.5. Hermitian Spaces 208 Chapter 6. Linear Operators 212 6.1. Matrix of a Linear Operator 212 6.2. Eigenvectors 218 6.3. Linear Operators and Bilinear Functions on Euclidean Space 223 6.4. Jordan Canonical Form 232 6.5. Functions of a Linear Operator 239 Chapter 7. Affine and Projective Spaces 250 7.1. Affine Spaces 250 7.2. Convex Sets 258 7.3. Affine Transformations and Motions 270 7.4. Quadrics 279 7.5. Projective Spaces 291 Chapter 8. Tensor Algebra 306 8.1. Tensor Product of Vector Spaces 306 8.2. Tensor Algebra of a Vector Space 313 8.3. Symmetric Algebra 319 8.4. Grassmann Algebra 325 Chapter 9. Commutative Algebra 336 9.1. Abelian Groups 336 9.2. Ideals and Quotient Rings 348 9.3. Modules over Principal Ideal Domains 356 9.4. Noetherian Rings 363 9.5. Algebraic Extensions 367 9.6. Finitely Generated Algebras and Affine Algebraic Varieties 378 9.7. Prime Factorization 387 Chapter 10. Groups 396 10.1. Direct and Semidirect Products 396 10.2. Commutator Subgroup 403 10.3. Group Actions 405 10.4. Sylow Theorems 411 10.5. Simple Groups 414 10.6. Galois Extensions 418 10.7. Fundamental Theorem of Galois Theory 423 Chapter 11. Linear Representations and Associative Algebras 430 11.1. Invariant Subspaces 430 11.2. Complete Reducibility of Linear Representations of Finite and Compact Groups 441 11.3. Finite-Dimensional Associative Algebras 445 11.4. Linear Representations of Finite Groups 453 11.5. Invariants 463 11.6. Division Algebras 469 Chapter 12. Lie Groups 482 12.1. Definition and Simple Properties of Lie Groups 483 12.2. The Exponential Map 489 12.3. Tangent Lie Algebra and the Adjoint Representation 493 12.4. Linear Representations of Lie Groups 498 Answers to Selected Exercises 506 Bibliography 512 Index 514 0821833189,9780821833186,4821834134 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 Chapter 1. Algebraic Structures Chapter 2. Elements Of Linear Algebra Chapter 3. Elements Of Polynomial Algebra Chapter 4. Elements Of Group Theory Chapter 5. Vector Spaces Chapter 6. Linear Operators Chapter 7. Affine And Projective Spaces Chapter 8. Tensor Algebra Chapter 9. Commutative Algebra Chapter 10. Groups Chapter 11. Linear Representations And Associative Algebras Chapter 12. Lie Groups Answers To Selected Exercises E.b. Vinberg. Includes Bibliographical References (p. 501-502) And Index.