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ISBN13: 978-1577664437

ISBN10: 1577664434

Edition: 3RD 06

Copyright: 2006

Publisher: Waveland Press, Inc.

Published: 2006

International: No

ISBN10: 1577664434

Edition: 3RD 06

Copyright: 2006

Publisher: Waveland Press, Inc.

Published: 2006

International: No

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Highly regarded by instructors in past editions for its sequencing of topics as well as its concrete approach, slightly slower beginning pace, and extensive set of exercises, the latest edition of Abstract Algebra extends the thrust of the widely used earlier editions as it introduces modern abstract concepts only after a careful study of important examples. Beachy and Blair's clear narrative presentation responds to the needs of inexperienced students who stumble over proof writing, who understand definitions and theorems but cannot do the problems, and who want more examples that tie into their previous experience. The authors introduce chapters by indicating why the material is important and, at the same time, relating the new material to things from the student's background and linking the subject matter of the chapter to the broader picture.

Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book. Rather than inserting superficial applications at the expense of important mathematical concepts, the Beachy and Blair solid, well-organized treatment motivates the subject with concrete problems from areas that students have previously encountered, namely, the integers and polynomials over the real numbers.

**1. Integers**

Divisors

Primes

Congruences

Integers Modulo n

**2. Functions**

Functions

Equivalence Relations

Permutations

**3. Groups**

Definition of a Group

Subgroups

Constructing Examples

Isomorphisms

Cyclic Groups

Permutation Groups

Homomorphisms

Cosets, Normal Subgroups, and Factor Groups

**4. Polynomials**

Fields; Roots of Polynomials

Factors

Existence of Roots

Polynomials over Z, Q, R, and C

**5. Commutative Rings**

Commutative Rings; Integral Domains

Ring Homomorphisms

Ideals and Factor Rings

Quotient Fields

**6. Fields**

Algebraic Elements

Finite and Algebraic Extensions

Geometric Constructions

Splitting Fields

Finite Fields

Irreducible Polynomials over Finite Fields

Quadratic Reciprocity

**7. Structure of Groups**

Isomorphism Theorems; Automorphisms

Conjugacy

Groups Acting on Sets

The Sylow Theorems

Finite Abelian Groups

Solvable Groups

Simple Groups

**8. Galois Theory**

The Galois Group of a Polynomial

Multiplicity of Roots

The Fundamental Theorem of Galois Theory

Solvability by Radicals

Cyclotomic Polynomials

Computing Galois Groups

**9. Unique Factorization**

Principal Ideal Domains

Unique Factorization Domains

Some Diophantine Equations

**Appendix**

Sets

Construction of the Number Systems

Basic Properties of the Integers

Induction

Complex Numbers

Solution of Cubic and Quartic Equations

Dimension of a Vector Space

John A. Beachy and William D. Blair

ISBN13: 978-1577664437ISBN10: 1577664434

Edition: 3RD 06

Copyright: 2006

Publisher: Waveland Press, Inc.

Published: 2006

International: No

Highly regarded by instructors in past editions for its sequencing of topics as well as its concrete approach, slightly slower beginning pace, and extensive set of exercises, the latest edition of Abstract Algebra extends the thrust of the widely used earlier editions as it introduces modern abstract concepts only after a careful study of important examples. Beachy and Blair's clear narrative presentation responds to the needs of inexperienced students who stumble over proof writing, who understand definitions and theorems but cannot do the problems, and who want more examples that tie into their previous experience. The authors introduce chapters by indicating why the material is important and, at the same time, relating the new material to things from the student's background and linking the subject matter of the chapter to the broader picture.

Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book. Rather than inserting superficial applications at the expense of important mathematical concepts, the Beachy and Blair solid, well-organized treatment motivates the subject with concrete problems from areas that students have previously encountered, namely, the integers and polynomials over the real numbers.

Table of Contents

**1. Integers**

Divisors

Primes

Congruences

Integers Modulo n

**2. Functions**

Functions

Equivalence Relations

Permutations

**3. Groups**

Definition of a Group

Subgroups

Constructing Examples

Isomorphisms

Cyclic Groups

Permutation Groups

Homomorphisms

Cosets, Normal Subgroups, and Factor Groups

**4. Polynomials**

Fields; Roots of Polynomials

Factors

Existence of Roots

Polynomials over Z, Q, R, and C

**5. Commutative Rings**

Commutative Rings; Integral Domains

Ring Homomorphisms

Ideals and Factor Rings

Quotient Fields

**6. Fields**

Algebraic Elements

Finite and Algebraic Extensions

Geometric Constructions

Splitting Fields

Finite Fields

Irreducible Polynomials over Finite Fields

Quadratic Reciprocity

**7. Structure of Groups**

Isomorphism Theorems; Automorphisms

Conjugacy

Groups Acting on Sets

The Sylow Theorems

Finite Abelian Groups

Solvable Groups

Simple Groups

**8. Galois Theory**

The Galois Group of a Polynomial

Multiplicity of Roots

The Fundamental Theorem of Galois Theory

Solvability by Radicals

Cyclotomic Polynomials

Computing Galois Groups

**9. Unique Factorization**

Principal Ideal Domains

Unique Factorization Domains

Some Diophantine Equations

**Appendix**

Sets

Construction of the Number Systems

Basic Properties of the Integers

Induction

Complex Numbers

Solution of Cubic and Quartic Equations

Dimension of a Vector Space

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