Algebra and Number Theory: An Integrated Approach

This book successfully blends algebra and number theory as an integrated discipline and consists of seven parts: Part 1 discusses the elements of set theory; Parts 2 and 3 address number systems; Parts 4 and 5 cover the main topics of linear algebra; Part 6 develops the main ideas of algebraic structures; and Part 7 demonstrates the applications of algebraic ideas to number theory.  Based on the experience of the authors, this book was developed for one course that integrates three disciplines – linear algebra, abstract algebra, and number theory – in an effort to use time more efficiently.  Many theorems in number theory have very simple proofs using algebraic tools, and most importantly, the book′s integrated approach helps to build a deeper understanding of the subject for readers as well as improve their retention of knowledge.  Applications are provided at the end of each chapter to further explain the results found in the book, and exercises are also ample throughout.  While the book is mathematically self–contained, readers should be comfortable with mathematical formalism and have some experience in reading and writing mathematical proofs. 

Spis treści:
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 Operat ions.
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.