## 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 Infinite-Dimensional 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 *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. 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 Skew-Symmetric 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 One-Parameter 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 One-Dimensional 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