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.

### Table of Contents

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.