Algebra and number theory: an integrated approach

Preface

Chapter 1. Sets

1.1. Operations on Sets

Exercise Set 1.1

1.2. Set Mappings

Exercise Set 1.2

1.3. Products of Mappings

Exercise Set 1.3

1.4. Some Properties of Integers

Exercise Set 1.4

Chapter 2. Matrices and Determinants

2.1. Operations on Matrices

Exercise Set 2.1

2.2. Permutations of Finite Sets

Exercise Set 2.2

2.3. Determinants of Matrices

Exercise Set 2.3

2.4. Computing Determinants

Exercise Set 2.4

2.5. Properties of the Product of Matrices

Exercise Set 2.5

Chapter 3. Fields

3.1. Binary Algebraic Operations

Exercise Set 3.1

3.2. Basic Properties of Fields

Exercise Set 3.2

3.3. The Field of Complex Numbers

Exercise Set 3.3

Chapter 4. Vector Spaces

4.1. Vector Spaces

Exercise Set 4.1

4.2. Dimension

Exercise Set 4.2

4.3. The Rank of a Matrix

Exercise Set 4.3

4.4. Quotient Spaces

Exercise Set 4.4

Chapter 5. Linear Mappings

5.1. Linear Mappings

Exercise Set 5.1

5.2. Matrices of Linear Mappings

Exercise Set 5.2

5.3. Systems of Linear Equations

Exercise Set 5.3

5.4. Eigenvectors and Eigenvalues

Exercise Set 5.4

Chapter 6. Bilinear Forms

6.1. Bilinear Forms

Exercise Set 6.1

6.2. Classical Forms

Exercise Set 6.2

6.3. Symmetric Forms over R

Exercise Set 6.3

6.4. Euclidean Spaces

Exercise Set 6.4

Chapter 7. Rings

7.1. Rings, Subrings, and Examples

Exercise Set 7.1

7.2. Equivalence Relations

Exercise Set 7.2

7.3. Ideals and Quotient Rings

Exercise Set 7.3

7.4. Homomorphisms of Rings

Exercise Set 7.4

7.5. Rings of Polynomials and Formal Power Series

Exercise Set 7.5

7.6. Rings of Multivariable Polynomials

Exercise Set 7.6

Chapter 8. Groups

8.1. Groups and Subgroups

Exercise Set 8.1

8.2. Examples of Groups and Subgroups

Exercise Set 8.2

8.3. Cosets

Exercise Set 8.3

8.4. Normal Subgroups and Factor Groups

Exercise Set 8.4

8.5. Homomorphisms of Groups

Exercise Set 8.5

Chapter 9. Arithmetic Properties of Rings

9.1. Extending Arithmetic to Commutative Rings

Exercise Set 9.1

9.2. Euclidean Rings

Exercise Set 9.2

9.3. Irreducible Polynomials

Exercise Set 9.3

9.4. Arithmetic Functions

Exercise Set 9.4

9.5. Congruences

Exercise Set 9.5

Chapter 10. The Real Number System

10.1. The Natural Numbers

10.2. The Integers

10.3. The Rationals

10.4. The Real Numbers

Answers to Selected Exercises

Index