ISBN13: 978-0534402648

ISBN10: 053440264X

Cover type:

Edition/Copyright: 6TH 05

Publisher: Brooks/Cole Publishing Co.

Published: 2005

International: No

ISBN10: 053440264X

Cover type:

Edition/Copyright: 6TH 05

Publisher: Brooks/Cole Publishing Co.

Published: 2005

International: No

Helping to make the study of modern algebra more accessible, this text gradually introduces and develops concepts through helpful features that provide guidance on the techniques of proof construction and logic analysis. The text develops mathematical maturity for students by presenting the material in a theorem-proof format, with definitions and major results easily located through a user-friendly format. The treatment is rigorous and self-contained, in keeping with the objectives of training the student in the techniques of algebra and of providing a bridge to higher-level mathematical courses.

**Benefits:**

- NEW! The text and examples have been carefully refined for clarity and ease of understanding.
- Biographical sketches of mathematicians who have influenced the development of the material are included to provide insight into the historical development of mathematics.
- A summary of key words and phrases at the end of each chapter provides a valuable overview of the chapter.
- An abundance of examples develop students' intuition.
- A list of special notations used in the book and the group tables for the most common examples are on the endpapers for quick and easy reference.
- NEW! Homework problems have been added, revised, and improved.
- NEW! The coverage of binary operations and permutations and inverses has been split into two sections in Chapter 1.
- NEW! An optional section on the solution of cubic and quartic equations by formulas appears in Chapter 8.
- Descriptive labels and titles are placed on definitions and theorems to indicate their content and relevance.
- Strategy boxes are included to give guidance and explanation about techniques of proof. This feature forms a component of the bridge that enables students to become more proficient in their poof construction skills.
- Symbolic marginal notes are used to help students analyze the logic in the proofs of theorems without interrupting the natural flow of the proof.
- A reference system provides guideposts to continuations and interconnections of exercises throughout the text.
- An appendix on the basics of logic and methods of proof is included to assist students with a weak background in logic.

1. FUNDAMENTALS.

Sets. Mappings. Properties of Composite Mappings (Optional). Binary Operations. Permutations and Inverses. Matrices. Relations. Key Words and Phrases. A Pioneer in Mathematics: Arthur Cayley.

2. THE INTEGERS.

Postulates for the Integers (Optional). Mathematical Induction. Divisibility. Prime Factors and Greatest Common Divisor. Congruence of Integers. Congruence Classes. Introduction to Coding Theory (Optional). Introduction to Cryptography (Optional). Key Words and Phrases. A Pioneer in Mathematics: Blaise Pascal.

3. GROUPS.

Definition of a Group. Subgroups. Cyclic Groups. Isomorphisms. Homomorphisms. Key Words and Phrases. A Pioneer in Mathematics: Niels Henrik Abel.

4. MORE ON GROUPS.

Finite Permutation Groups. Cayley's Theorem. Permutation Groups in Science and Art (Optional). Normal Subgroups. Quotient Groups. Direct Sums (Optional). Some Results on Finite Abelian Groups (Optional). Key Words and Phrases. A Pioneer in Mathematics: Augustin Louis Cauchy.

5. RINGS, INTEGRAL DOMAINS, AND FIELDS.

Definition of a Ring. Integral Domains and Fields. The Field of Quotients of an Integral Domain. Ordered Integral Domains. Key Words and Phrases. A Pioneer in Mathematics: Richard Dedekind.

6. MORE ON RINGS.

Ideals and Quotient Rings. Ring Homomorphisms. The Characteristic of a Ring. Maximal Ideals (Optional). Key Words and Phrases. A Pioneer in Mathematics: Amalie Emmy Noether.

7. REAL AND COMPLEX NUMBERS.

The Field of Real Numbers. Complex Numbers and Quaternions. De Moivre's Theorem and Roots of Complex Numbers. Key Words and Phrases. A Pioneer in Mathematics: William Rowan Hamilton.

8. POLYNOMIALS.

Polynomials over a Ring. Divisibility and Greatest Common Divisor. Factorization in F[x]. Zeros of a Polynomial. Solutions of Cubic and Quartic Equations by Formulas (Optional). Algebraic Extensions of a Field. Key Words and Phrases. A Pioneer in Mathematics: Carl Friedrich Gauss.

Appendix: The Basics Of Logic.

Bibliography.

Answers to Selected Computational Exercises.

Index.

Jimmie Gilbert and Linda Gilbert

ISBN13: 978-0534402648ISBN10: 053440264X

Cover type:

Edition/Copyright: 6TH 05

Publisher: Brooks/Cole Publishing Co.

Published: 2005

International: No

Helping to make the study of modern algebra more accessible, this text gradually introduces and develops concepts through helpful features that provide guidance on the techniques of proof construction and logic analysis. The text develops mathematical maturity for students by presenting the material in a theorem-proof format, with definitions and major results easily located through a user-friendly format. The treatment is rigorous and self-contained, in keeping with the objectives of training the student in the techniques of algebra and of providing a bridge to higher-level mathematical courses.

**Benefits:**

- NEW! The text and examples have been carefully refined for clarity and ease of understanding.
- Biographical sketches of mathematicians who have influenced the development of the material are included to provide insight into the historical development of mathematics.
- A summary of key words and phrases at the end of each chapter provides a valuable overview of the chapter.
- An abundance of examples develop students' intuition.
- A list of special notations used in the book and the group tables for the most common examples are on the endpapers for quick and easy reference.
- NEW! Homework problems have been added, revised, and improved.
- NEW! The coverage of binary operations and permutations and inverses has been split into two sections in Chapter 1.
- NEW! An optional section on the solution of cubic and quartic equations by formulas appears in Chapter 8.
- Descriptive labels and titles are placed on definitions and theorems to indicate their content and relevance.
- Strategy boxes are included to give guidance and explanation about techniques of proof. This feature forms a component of the bridge that enables students to become more proficient in their poof construction skills.
- Symbolic marginal notes are used to help students analyze the logic in the proofs of theorems without interrupting the natural flow of the proof.
- A reference system provides guideposts to continuations and interconnections of exercises throughout the text.
- An appendix on the basics of logic and methods of proof is included to assist students with a weak background in logic.

Table of Contents

1. FUNDAMENTALS.

Sets. Mappings. Properties of Composite Mappings (Optional). Binary Operations. Permutations and Inverses. Matrices. Relations. Key Words and Phrases. A Pioneer in Mathematics: Arthur Cayley.

2. THE INTEGERS.

Postulates for the Integers (Optional). Mathematical Induction. Divisibility. Prime Factors and Greatest Common Divisor. Congruence of Integers. Congruence Classes. Introduction to Coding Theory (Optional). Introduction to Cryptography (Optional). Key Words and Phrases. A Pioneer in Mathematics: Blaise Pascal.

3. GROUPS.

Definition of a Group. Subgroups. Cyclic Groups. Isomorphisms. Homomorphisms. Key Words and Phrases. A Pioneer in Mathematics: Niels Henrik Abel.

4. MORE ON GROUPS.

Finite Permutation Groups. Cayley's Theorem. Permutation Groups in Science and Art (Optional). Normal Subgroups. Quotient Groups. Direct Sums (Optional). Some Results on Finite Abelian Groups (Optional). Key Words and Phrases. A Pioneer in Mathematics: Augustin Louis Cauchy.

5. RINGS, INTEGRAL DOMAINS, AND FIELDS.

Definition of a Ring. Integral Domains and Fields. The Field of Quotients of an Integral Domain. Ordered Integral Domains. Key Words and Phrases. A Pioneer in Mathematics: Richard Dedekind.

6. MORE ON RINGS.

Ideals and Quotient Rings. Ring Homomorphisms. The Characteristic of a Ring. Maximal Ideals (Optional). Key Words and Phrases. A Pioneer in Mathematics: Amalie Emmy Noether.

7. REAL AND COMPLEX NUMBERS.

The Field of Real Numbers. Complex Numbers and Quaternions. De Moivre's Theorem and Roots of Complex Numbers. Key Words and Phrases. A Pioneer in Mathematics: William Rowan Hamilton.

8. POLYNOMIALS.

Polynomials over a Ring. Divisibility and Greatest Common Divisor. Factorization in F[x]. Zeros of a Polynomial. Solutions of Cubic and Quartic Equations by Formulas (Optional). Algebraic Extensions of a Field. Key Words and Phrases. A Pioneer in Mathematics: Carl Friedrich Gauss.

Appendix: The Basics Of Logic.

Bibliography.

Answers to Selected Computational Exercises.

Index.

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