A course in Algebra

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. 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 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.