Table of Contents

• 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

• 2.1 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

• 3.1 Subspaces of Rⁿ
• 3.2 Fields
• 3.3 Vector Spaces
• 3.4 Bases and Dimension
• 3.5 Computing with Bases
• 3.6 Direct Sums
• 3.7 Infinite-Dimensional Spaces
• 3.8 Exercises

• 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

• 5.1 Orthogonal Matrices and Rotations
• 5.2 Using Continuity
• 5.3 Systems of Differential Equations
• 5.4 The Matrix Exponential
• 5.5 Exercises

• 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

• 7.1 Cayley's Theorem
• 7.2 The Class Equation
• 7.3 r-groups
• 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 Todd-Coxeter Algorithm
• 7.12 Exercises

• 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 Skew-Symmetric Forms
• 8.9 Summary
• 8.10 Exercises

• 9.1 The Classical Groups
• 9.2 Interlude: Spheres
• 9.3 The Special Unitary Group SU₂
• 9.4 The Rotation Group SO₃
• 9.5 One-Parameter Groups
• 9.6 The Lie Algebra
• 9.7 Translation in a Group
• 9.8 Normal Subgroups of SL₂
• 9.9 Exercises

• 10.1 Definitions
• 10.2 Irreducible Representations
• 10.3 Unitary Representations
• 10.4 Characters
• 10.5 One-Dimensional Characters
• 10.6 The Regular Representations
• 10.7 Schur's Lemma
• 10.8 Proof of the Orthogonality Relations
• 10.9 Representationsof SU₂
• 10.10 Exercises

• 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

• 12.1 Factoring Integers
• 12.2 Unique Factorization Domains
• 12.3 Gauss's Lemma
• 12.4 Factoring Integer Polynomial
• 12.5 Gauss Primes
• 12.6 Exercises

• 13.1 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

• 14.1 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

• 15.1 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

• 16.1 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

• A.1 About Proofs
• A.2 The Integers
• A.3 Zorn's Lemma
• A.4 The Implicit Function Theorem
• A.5 Exercises