A Course in Algebra 1st Edition by Ernest Borisovich Vinberg ISBN 0821833189 978-0821833186
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ISBN 10: 0821833189
ISBN 13: 978-0821833186
Author: Ernest Borisovich Vinberg
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.
Table of contents:
Chapter 1. Algebraic Structures
§1.1. Introduction
§1.2. Abelian Groups
§1.3. Rings and Fields
§1.4. Subgroups, Subrings, and Subfields
§1.5. The Field of Complex Numbers
§1.6. Rings of Residue Classes
§1.7. Vector Spaces
§1.8. Algebras
§1.9. Matrix Algebras
Chapter 2. Elements of Linear Algebra
§2.1. Systems of Linear Equations
§2.2. Basis and Dimension of a Vector Space
§2.3. Linear Maps
§2.4. Determinants
§2.5. Several Applications of Determinants
Chapter 3. Elements of Polynomial Algebra
§3.1. Polynomial Algebra: Construction and Basic Properties
§3.2. Roots of Polynomials: General Properties
§3.3. Fundamental Theorem of Algebra of Complex Numbers
§3.4. Roots of Polynomials with Real Coefficients
§3.5. Factorization in Euclidean Domains
§3.6. Polynomials with Rational Coefficients
§3.7. Polynomials in Several Variables
§3.8. Symmetric Polynomials
§3.9. Cubic Equations
§3.10. Field of Rational Fractions
Chapter 4. Elements of Group Theory
§4.1. Definitions and Examples
§4.2. Groups in Geometry and Physics
§4.3. Cyclic Groups
§4.4. Generating Sets
§4.5. Cosets
§4.6. Homomorphisms
Chapter 5. Vector Spaces
§5.1. Relative Position of Subspaces
§5.2. Linear Functions
§5.3. Bilinear and Quadratic Functions
§5.4. Euclidean Spaces
§5.5. Hermitian Spaces
Chapter 6. Linear Operators
§6.1. Matrix of a Linear Operator
§6.2. Eigenvectors
§6.3. Linear Operators and Bilinear Functions on Euclidean Space
§6.4. Jordan Canonical Form
§6.5. Functions of a Linear Operator
Chapter 7. Affine and Projective Spaces
§7.1. Affine Spaces
§7.2. Convex Sets
§7.3. Affine Transformations and Motions
§7.4. Quadrics
§7.5. Projective Spaces
Chapter 8. Tensor Algebra
§8.1. Tensor Product of Vector Spaces
§8.2. Tensor Algebra of a Vector Space
§8.3. Symmetric Algebra
§8.4. Grassmann Algebra
Chapter 9. Commutative Algebra
§9.1. Abelian Groups
§9.2. Ideals and Quotient Rings
§9.3. Modules over Principal Ideal Domains
§9.4. Noetherian Rings
§9.5. Algebraic Extensions
§9.6. Finitely Generated Algebras and Affine Algebraic Varieties
§9.7. Prime Factorization
Chapter 10. Groups
§10.1. Direct and Semidirect Products
§10.2. Commutator Subgroup
§10.3. Group Actions
§10.4. Sylow Theorems
§10.5. Simple Groups
§10.6. Galois Extensions
§10.7. Fundamental Theorem of Galois Theory
Chapter 11. Linear Representations and Associative Algebras
§11.1. Invariant Subspaces
§11.2. Complete Reducibility of Linear Representations of Finite and Compact Groups
§11.3. Finite-Dimensional Associative Algebras
§11.4. Linear Representations of Finite Groups
§11.5. Invariants
§11.6. Division Algebras
Chapter 12. Lie Groups
§12.1. Definition and Simple Properties of Lie Groups
§12.2. The Exponential Map
§12.3. Tangent Lie Algebra and the Adjoint Representation
§12.4. Linear Representations of Lie Groups
Answers to Selected Exercises
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