Abstract Algebra

Description

• provides ample foundational material—both at the beginning of the text and in the appendices—while not avoiding calculus.

• features a “rings” first approach to make abstract concepts more accessible. Over 225 substantial examples and 750 exercises (many having multiple parts) are provided.

• Geometry is slowly integrated through the text to make the abstract concrete and permit more structures to play with.

• culminates in Chapter 8 with the true integration of geometry and algebra.

• encourages the use of geometric models to discover the relations between group theory and geometry.

This book explores the essential theories and techniques of modern algebra, including its problem-solving skills, basic proof techniques, many unusual applications, and the interplay between algebra and geometry. It takes a concrete, example-oriented approach to the subject matter.

1. The Integers.

2. From the Integers to the Complex Numbers.

3. Polynomials.

4. Homomorphisms and Quotient Rings.

5. Field Extensions.

6. Groups.

7. Group Actions and Symmetry.

8. Non-Euclidean Geometries.

Appendix A. A Logic Review, Sets and Functions, and Equivalence Relations.

Appendix B. Miscellaneous Facts from Linear Algebra.

Supplementary Reading.

Table of Notations.

Index.

Additional information