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.
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 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.
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.
* Applications of Grobner bases to computations in algebraic geometry in Chapter 15
* Construction of the simple group of order 168 using the projective plane of order 2 (the Fano plane)
- Accessible to undergraduates yet its scope and depth also make it ideal for courses at the graduate level.
- Over 1950 exercises, many with multiple parts, ranging in scope from routine to fairly sophisticated, and ranging in purpose from basic application of text material to exploration of important theoretical or computational techniques.
- The structure of the book permits instructors and students to pursue certain areas from their beginnings to an in-depth treatment, or to survey a wider range of areas, seeing how various themes recur and how different structures are related.
- The emphasis throughout is to motivate the introduction and development of important algebraic concepts using as many examples as possible. The wealth of examples helps to ground the theory, explain its application, and help develop the student's intuition.
- Contains many topics not usually found in introductory texts. Students are able to see how these fit naturally into the main themes of algebra. Examples of these topics include:
- Rings of algebraic integers
- Semidirect products and the theory of extensions
- Criteria for Principal Ideal Domains
- Criteria for the solvability of a quintic
- Dedekind Domains
- Algorithm for determining whether an ideal in a polynomial ring over a field is a prime ideal
- Affine schemes
- Galois cohomology
- Burnside's Theorem on the solvability of groups of order paqb