Table of Contents
by David S. Dummit, Richard M. Foote
This revision of Dummit and Foote’s widely acclaimed introduction to abstract algebra helps students experience the power and beauty that develops from the rich interplay between different areas of mathematics.
This book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the student’s understanding. With this approach, students gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings.
The text is designed for a full-year introduction to abstract algebra at the advanced undergraduate or graduate level but contains substantially more material than would normally be covered in one year. Portions of the book may also be used for various one-semester topics courses in advanced algebra, each of which would provide a solid background for a follow-up course delving more deeply into one of many possible areas: algebraic number theory, algebraic topology, algebraic geometry, representation theory, Lie groups, etc.
This book is designed to give the reader insight into the power and beauty that accrues from a rich interplay between different areas of mathematics. The book carefully develops the theory of different algebraic structures, beginning from basic definitions to some in-depth results, using numerous examples and exercises to aid the reader’s understanding. In this way, readers gain an appreciation for how mathematical structures and their interplay lead to powerful results and insights in a number of different settings.
Table of Contents
PART I: GROUP THEORY.
Chapter 1. Introduction to Groups.
Chapter 2. Subgroups.
Chapter 3. Quotient Group and Homomorphisms.
Chapter 4. Group Actions.
Chapter 5. Direct and Semidirect Products and Abelian Groups.
Chapter 6. Further Topics in Group Theory.
PART II: RING THEORY.
Chapter 7. Introduction to Rings.
Chapter 8. Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains.
Chapter 9. Polynomial Rings.
PART III: MODULES AND VECTOR SPACES.
Chapter 10. Introduction to Module Theory.
Chapter 11. Vector Spaces.
Chapter 12. Modules over Principal Ideal Domains.
PART IV: FIELD THEORY AND GALOIS THEORY.
Chapter 13. Field Theory.
Chapter 14. Galois Theory.
PART V: AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOMETRY, AND HOMOLOGICAL ALGEBRA.
Chapter 15. Commutative Rings and Algebraic Geometry.
Chapter 16. Artinian Rings, Discrete Valuation Rings, and Dedekind Domains.
Chapter 17. Introduction to Homological Algebra and Group Cohomology.
PART VI: INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS.
Chapter 18. Representation Theory and Character Theory.
Chapter 19. Examples and Applications of Character Theory.
Appendix I: Cartesian Products and Zorn’s Lemma.
Appendix II: Category Theory.
Some Related Pages