Elements of Modern Algebra,
8th Edition

1. FUNDAMENTALS.
Sets. Mappings. Properties of Composite Mappings (Optional). Binary Operations. Permutations and Inverses. Matrices. Relations.
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).
3. GROUPS.
Definition of a Group. Properties of Group Elements. Subgroups. Cyclic Groups. Isomorphisms. Homomorphisms.
4. MORE ON GROUPS.
Finite Permutation Groups. Cayley's Theorem. Permutation Groups in Science and Art (Optional). Cosets of a Subgroup. Normal Subgroups. Quotient Groups. Direct Sums (Optional). Some Results on Finite Abelian Groups (Optional).
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.
6. MORE ON RINGS.
Ideals and Quotient Rings. Ring Homomorphisms. The Characteristic of a Ring. Maximal Ideals (Optional).
7. REAL AND COMPLEX NUMBERS.
The Field of Real Numbers. Complex Numbers and Quaternions. De Moivre's Theorem and Roots of Complex Numbers.
8. POLYNOMIALS.
Polynomials over a Ring. Divisibility and Greatest Common Divisor. Factorization in _F[x]_ . Zeros of a Polynomial. Solution of Cubic and Quartic Equations by Formulas (Optional). Algebraic Extensions of a Field.

• Linda Gilbert

  Linda Gilbert received her Ph.D. from Louisiana Tech University with a specialty in Linear and Abstract Algebras. She has been writing textbooks since 1981 with her husband Jimmie Gilbert, including ELEMENTS OF MODERN ALGEBRA and LINEAR ALGEBRA and MATRIX THEORY (now in its second edition) with Cengage Learning, plus titles in College Algebra, Precalculus, College Algebra and Trigonometry, Trigonometry, and Intermediate Algebra.

• Alerts that draw attention to counterexamples, special cases, proper symbol or terminology usage, and common misconceptions. Frequently these alerts lead to True/False statements in the exercises that further reinforce the precision required in mathematical communication.

• More emphasis placed on special groups, such as the general linear and special linear groups, the dihedral group, and the group of units.

• Moving some definitions from the exercises to the sections for greater emphasis.

• Using marginal notes to outline the steps of the induction arguments required in the examples.

• More than 200 new theoretical and computational exercises have been added.

• Nearly 300 True/False statements that encourage the students to thoroughly understand the statements of definitions and results of theorems appear in this edition.

• Descriptive labels and titles are used with definitions and theorems to indicate their content and relevance.

• Strategy boxes appear 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 proof 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.

• Biographical sketches of great mathematicians whose contributions are relevant to the respective material conclude each chapter.

• A summary of key words and phrases is included at the end of each chapter.

• A list of special notations used in the book appears on the front endpapers.

• Group tables for the most common examples are on the back endpapers.

• An updated bibliography is included.

• Strategy boxes appear 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 proof construction skills.

• A reference system provides guideposts to continuations and interconnections of exercises throughout the text.

• Symbolic marginal notes are used to help students analyze the logic in the proofs of theorems without interrupting the natural flow of the proof.

Cengage provides a range of supplements that are updated in coordination with the main title selection. For more information about these supplements, contact your Learning Consultant.

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