Algebra (Classic Version), 2nd edition
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Overview
Hallmark features of this title
 High emphasis on concrete topics such as symmetry, linear groups, quadratic number fields, and lattices prepares students to learn more abstract concepts. This focus also allows some abstractions to be treated more concisely.
 The chapter organization emphasizes the connections between algebra and geometry at the start, with the beginning chapters containing the content most important for students in other fields. To counter the fact that arithmetic receives less initial emphasis, the later chapters have a strong arithmetic slant.
 Treatment beyond the basics sets this book apart. Students with a reasonably mature mathematical background will benefit from the relatively informal treatments the author gives to the more advanced topics.
 Content notes in the preface include teaching tips from the author’s own classroom experience.
 Challenging exercises are indicated with an asterisk, allowing instructors to easily create the right assignments for their class.
Published by Pearson (June 1st 2023)  Copyright © 2023
ISBN13: 9780137980994
Subject: Advanced Math
Category: Abstract Algebra
Table of contents
1. Matrices
 1.1 The Basic Operations
 1.2 Row Reduction
 1.3 The Matrix Transpose
 1.4 Determinants
 1.5 Permutations
 1.6 Other Formulas for the Determinant
 1.7 Exercises
2. Groups
 2.1 Laws of Composition
 2.2 Groups and Subgroups
 2.3 Subgroups of the Additive Group of Integers
 2.4 Cyclic Groups
 2.5 Homomorphisms
 2.6 Isomorphisms
 2.7 Equivalence Relations and Partitions
 2.8 Cosets
 2.9 Modular Arithmetic
 2.10 The Correspondence Theorem
 2.11 Product Groups
 2.12 Quotient Groups
 2.13 Exercises
3. Vector Spaces
 3.1 Subspaces of R^{n}
 3.2 Fields
 3.3 Vector Spaces
 3.4 Bases and Dimension
 3.5 Computing with Bases
 3.6 Direct Sums
 3.7 InfiniteDimensional Spaces
 3.8 Exercises
4. Linear Operators
 4.1 The Dimension Formula
 4.2 The Matrix of a Linear Transformation
 4.3 Linear Operators
 4.4 Eigenvectors
 4.5 The Characteristic Polynomial
 4.6 Triangular and Diagonal Forms
 4.7 Jordan Form
 4.8 Exercises
5. Applications of Linear Operators
 5.1 Orthogonal Matrices and Rotations
 5.2 Using Continuity
 5.3 Systems of Differential Equations
 5.4 The Matrix Exponential
 5.5 Exercises
6. Symmetry
 6.1 Symmetry of Plane Figures
 6.2 Isometries
 6.3 Isometries of the Plane
 6.4 Finite Groups of Orthogonal Operators on the Plane
 6.5 Discrete Groups of Isometries
 6.6 Plane Crystallographic Groups
 6.7 Abstract Symmetry: Group Operations
 6.8 The Operation on Cosets
 6.9 The Counting Formula
 6.10 Operations on Subsets
 6.11 Permutation Representation
 6.12 Finite Subgroups of the Rotation Group
 6.13 Exercises
7. More Group Theory
 7.1 Cayley's Theorem
 7.2 The Class Equation
 7.3 rgroups
 7.4 The Class Equation of the Icosahedral Group
 7.5 Conjugation in the Symmetric Group
 7.6 Normalizers
 7.7 The Sylow Theorems
 7.8 Groups of Order 12
 7.9 The Free Group
 7.10 Generators and Relations
 7.11 The ToddCoxeter Algorithm
 7.12 Exercises
8. Bilinear Forms
 8.1 Bilinear Forms
 8.2 Symmetric Forms
 8.3 Hermitian Forms
 8.4 Orthogonality
 8.5 Euclidean spaces and Hermitian spaces
 8.6 The Spectral Theorem
 8.7 Conics and Quadrics
 8.8 SkewSymmetric Forms
 8.9 Summary
 8.10 Exercises
9. Linear Groups
 9.1 The Classical Groups
 9.2 Interlude: Spheres
 9.3 The Special Unitary Group SU_{2}
 9.4 The Rotation Group SO_{3}
 9.5 OneParameter Groups
 9.6 The Lie Algebra
 9.7 Translation in a Group
 9.8 Normal Subgroups of SL_{2}
 9.9 Exercises
10. Group Representations
 10.1 Definitions
 10.2 Irreducible Representations
 10.3 Unitary Representations
 10.4 Characters
 10.5 OneDimensional Characters
 10.6 The Regular Representations
 10.7 Schur's Lemma
 10.8 Proof of the Orthogonality Relations
 10.9 Representationsof SU_{2}
 10.10 Exercises
11. Rings
 11.1 Definition of a Ring
 11.2 Polynomial Rings
 11.3 Homomorphisms and Ideals
 11.4 Quotient Rings
 11.5 Adjoining Elements
 11.6 Product Rings
 11.7 Fraction Fields
 11.8 Maximal Ideals
 11.9 Algebraic Geometry
 11.10 Exercises
12. Factoring
 12.1 Factoring Integers
 12.2 Unique Factorization Domains
 12.3 Gauss's Lemma
 12.4 Factoring Integer Polynomial
 12.5 Gauss Primes
 12.6 Exercises
13. Quadratic Number Fields
 13.1 Algebraic Integers
 13.2 Factoring Algebraic Integers
 13.3 Ideals in Z √(5)
 13.4 Ideal Multiplication
 13.5 Factoring Ideals
 13.6 Prime Ideals and Prime Integers
 13.7 Ideal Classes
 13.8 Computing the Class Group
 13.9 Real Quadratic Fields
 13.10 About Lattices
 13.11 Exercises
14. Linear Algebra in a Ring
 14.1 Modules
 14.2 Free Modules
 14.3 Identities
 14.4 Diagonalizing Integer Matrices
 14.5 Generators and Relations
 14.6 Noetherian Rings
 14.7 Structure to Abelian Groups
 14.8 Application to Linear Operators
 14.9 Polynomial Rings in Several Variables
 14.10 Exercises
15. Fields
 15.1 Examples of Fields
 15.2 Algebraic and Transcendental Elements
 15.3 The Degree of a Field Extension
 15.4 Finding the Irreducible Polynomial
 15.5 Ruler and Compass Constructions
 15.6 Adjoining Roots
 15.7 Finite Fields
 15.8 Primitive Elements
 15.9 Function Fields
 15.10 The Fundamental Theorem of Algebra
 15.11 Exercises
16. Galois Theory
 16.1 Symmetric Functions
 16.2 The Discriminant
 16.3 Splitting Fields
 16.4 Isomorphisms of Field Extensions
 16.5 Fixed Fields
 16.6 Galois Extensions
 16.7 The Main Theorem
 16.8 Cubic Equations
 16.9 Quartic Equations
 16.10 Roots of Unity
 16.11 Kummer Extensions
 16.12 Quintic Equations
 16.13 Exercises
Appendix A. Background Material
 A.1 About Proofs
 A.2 The Integers
 A.3 Zorn's Lemma
 A.4 The Implicit Function Theorem
 A.5 Exercises
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