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ELEMENTS OF MODERN ALGEBRA, Eighth Edition, is intended for an introductory course in abstract algebra taken by Math and Math for Secondary Education majors. Helping to make the study of abstract 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. The text has a flexible organization, with section dependencies clearly mapped out and optional topics that instructors can cover or skip based on their course needs. Additionally, problem sets are carefully arranged in order of difficulty to cater assignments to varying student ability levels.
- 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.
- Many new examples have also been added to this edition.
- 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.
- 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.
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).
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.
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.