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.
provides ample foundational materialboth at the beginning
of the text and in the appendiceswhile 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
encourages the use of geometric models to discover the relations
between group theory and geometry.
1. The Integers.
2. From the Integers to the Complex Numbers.
4. Homomorphisms and Quotient Rings.
5. Field Extensions.
7. Group Actions and Symmetry.
8. NonEuclidean Geometries.
Appendix A. A Logic Review, Sets and Functions, and Equivalence
Appendix B. Miscellaneous Facts from Linear Algebra.
Table of Notations.