## Reviews

A friendly introduction to the subject—no sentence is terse, and in fact useful additional statements are sprinkled throughout an argument... Suitably chosen examples are given throughout the text to illustrate the definitions, proofs and arguments, and there are also plenty of exercises at the end of each chapter for the reinforcement of understanding. It is an excellent text for a university course.

The utmost detailed presentation of the core material, the wealth of illustrating examples, and the many outlooks for further study make this excellent algebra primer a highly welcome, useful and valuable addition to the abundant textbook literature in the field.

## Book Details

Preface

1. Abstract Algebra and Algebraic Reasoning

1.1. Abstract Algebra

1.2. Algebraic Structures

1.3. The Algebraic Method

1.4. The Standard Number Systems

1.5. The Integers and Induction

1.6. Exercises

2

Preface

1. Abstract Algebra and Algebraic Reasoning

1.1. Abstract Algebra

1.2. Algebraic Structures

1.3. The Algebraic Method

1.4. The Standard Number Systems

1.5. The Integers and Induction

1.6. Exercises

2. Algebraic Preliminaries

2.1. Sets and Set Theory

2.1.1. Set Operations

2.2. Functions

2.3. Equivalence Relations and Factor Sets

2.4. Sizes of Sets

2.5. Binary Operations

2.5.1. The Algebra of Sets

2.6. Algebraic Structures and Isomorphisms

2.7. Groups

2.8. Exercises

3. Rings and the Integers

3.1. Rings and the Ring of Integers

3.2. Some Basic Properties of Rings and Subrings

3.3. Examples of Rings

3.3.1. The Modular Rings: The Integers Modulo n

3.3.2. Noncommutative Rings

3.3.3. Rings Without Identities

3.3.4. Rings of Subsets: Boolean Rings

3.3.5. Direct Sums of Rings

3.3.6. Summary of Examples

3.4. Ring Homomorphisms and Isomorphisms

3.5. Integral Domains and Ordering

3.6. Mathematical Induction and the Uniqueness of Z

3.7. Exercises

4. Number Theory and Unique Factorization

4.1. Elementary Number Theory

4.2. Divisibility and Primes

4.3. Greatest Common Divisors

4.4. The Fundamental Theorem of Arithmetic

4.5. Congruences and Modular Arithmetic

4.6. Unique Factorization Domains

4.7. Exercises

5. Fields: The Rationals, Reals and Complexes

5.1. Fields and Division Rings

5.2. Construction and Uniqueness of the Rationals

5.2.1. Fields of Fractions

5.3. The Real Number System

5.3.1. The Completeness of R (Optional)

5.3.2. Characterization of R (Optional)

5.3.3. The Construction of R (Optional)

5.3.4. The p-adic Numbers (Optional)

5.4. The Field of Complex Numbers

5.4.1. Geometric Interpretation

5.4.2. Polar Form and Euler's Identity

5.4.3. DeMoivre's Theorem for Powers and Roots

5.5. Exercises

6. Basic Group Theory

6.1. Groups, Subgroups and Isomorphisms

6.2. Examples of Groups

6.2.1. Permutations and the Symmetric Group

6.2.2. Examples of Groups: Geometric Transformation Groups

6.3. Subgroups and Lagrange's Theorem

6.4. Generators and Cyclic Groups

6.5. Exercises

7. Factor Groups and the Group Isomorphism Theorems

7.1. Normal Subgroups

7.2. Factor Groups

7.2.1. Examples of Factor Groups

7.3. The Group Isomorphism Theorems

7.4. Exercises

8. Direct Products and Abelian Groups

8.1. Direct Products of Groups

8.1.1. Direct Products of Two Groups

8.1.2. Direct Products of Any Finite Number of Groups

8.2. Abelian Groups

8.2.1. Finite Abelian Groups

8.2.2. Free Abelian Groups

8.2.3. The Basis Theorem for Finitely Generated Abelian Groups

8.3. Exercises

9. Symmetric and Alternating Groups

9.1. Symmetric Groups and Cycle Structure

9.1.1. The Alternating Groups

9.1.2. Conjugation in Sn

9.2. The Simplicity of An

9.3. Exercises

10. Group Actions and Topics in Group Theory

10.1. Group Actions

10.2. Conjugacy Classes and the Class Equation

10.3. The Sylow Theorems

10.3.1. Some Applications of the Sylow Theorems

10.4. Groups of Small Order

10.5. Solvability and Solvable Groups

10.5.1. Solvable Groups

10.5.2. The Derived Series

10.6. Composition Series and the Jordan-Holder Theorem

10.7. Exercises

11. Topics in Ring Theory

11.1. Ideals in Rings

11.2. Factor Rings and the Ring Isomorphism Theorem

11.3. Prime and Maximal Ideals

11.3.1. Prime Ideals and Integral Domains

11.3.2. Maximal Ideals and Fields

11.4. Principal Ideal Domains and Unique Factorization

11.5. Exercises

12. Polynomials and Polynomial Rings

12.1. Polynomials and Polynomial Rings

12.2. Polynomial Rings over a Field

12.2.1. Unique Factorization of Polynomials

12.2.2. Euclidean Domains

12.2.3. F[x] as a Principal Ideal Domain

12.2.4. Polynomial Rings over Integral Domains

12.3. Zeros of Polynomials

12.3.1. Real and Complex Polynomials

12.3.2. The Fundamental Theorem of Algebra

12.3.3. The Rational Roots Theorem

12.3.4. Solvability by Radicals

12.3.5. Algebraic and Transcendental Numbers

12.4. Unique Factorization in Z[x]

12.5. Exercises

13. Algebraic Linear Algebra

13.1. Linear Algebra

13.1.1. Vector Analysis in R3

13.1.2. Matrices and Matrix Algebra

13.1.3. Systems of Linear Equations

13.1.4. Determinants

13.2. Vector Spaces over a Field

13.2.1. Euclidean n-Space

13.2.2. Vector Spaces

13.2.3. Subspaces

13.2.4. Bases and Dimension

13.2.5. Testing for Bases in Fn

13.3. Dimension and Subspaces

13.4. Algebras

13.5. Inner Product Spaces

13.5.1. Banach and Hilbert Spaces

13.5.2. The Gram-Schmidt Process and Orthonormal Bases

13.5.3. The Closest Vector Theorem

13.5.4. Least-Squares Approximation

13.6. Linear Transformations and Matrices

13.6.1. Matrix of a Linear Transformation

13.6.2. Linear Operators and Linear Functionals

13.7. Exercises

14. Fields and Field Extensions

14.1. Abstract Algebra and Galois Theory

14.2. Field Extensions

14.3. Algebraic Field Extensions

14.4. F-automorphisms, Conjugates and Algebraic Closures

14.5. Adjoining Roots to Fields

14.6. Splitting Fields and Algebraic Closures

14.7. Automorphisms and Fixed Fields

14.8. Finite Fields

14.9. Transcendental Extensions

14.10. Exercises

15. A Survey of Galois Theory

15.1. An Overview of Galois Theory

15.2. Galois Extensions

15.3. Automorphisms and the Galois Group

15.4. The Fundamental Theorem of Galois Theory

15.5. A Proof of the Fundamental Theorem of Algebra

15.6. Some Applications of Galois Theory

15.6.1. The Insolvability of the Quintic

15.6.2. Some Ruler and Compass Constructions

15.6.3. Algebraic Extensions of R

15.7. Exercises

Bibliography

Index