As the author notes in the preface, "The purpose of this book is to acquaint a broad spectrum of students with what is today known as 'abstract algebra.'" Written for a one-semester course, this self-contained text includes numerous examples designed to base the definitions and theorems on experience, to illustrate the theory with concrete examples in familiar contexts, and to give the student extensive computational practice. The first three chapters progress in a relatively ...
As the author notes in the preface, "The purpose of this book is to acquaint a broad spectrum of students with what is today known as 'abstract algebra.'" Written for a one-semester course, this self-contained text includes numerous examples designed to base the definitions and theorems on experience, to illustrate the theory with concrete examples in familiar contexts, and to give the student extensive computational practice.
The first three chapters progress in a relatively leisurely fashion and include abundant detail to make them as comprehensible as possible. Chapter One provides a short course in sets and numbers for students lacking those prerequisites, rendering the book largely self-contained. While Chapters Four and Five are more challenging, they are well within the reach of the serious student.
The exercises have been carefully chosen for maximum usefulness. Some are formal and manipulative, illustrating the theory and helping to develop computational skills. Others constitute an integral part of the theory, by asking the student to supply proofs or parts of proofs omitted from the text. Still others stretch mathematical imaginations by calling for both conjectures and proofs.
Taken together, text and exercises comprise an excellent introduction to the power and elegance of abstract algebra. Now available in this inexpensive edition, the book is accessible to a wide range of students, who will find it an exceptionally valuable resource.
1. Sets and Numbers
I. THE ELEMENTS OF SET THEORY
1. The Concept of Set
2. "Constants, Variables and Related Matters"
3. Subsets and Equality of Sets
4. The Algebra of Sets; The Empty Set
5. A Notation for Sets
6. Generalized Intersection and Union
7. Ordered Pairs and Cartesian Products
8. Functions (or Mappings)
9. A Classification of Mappings
10. Composition of Mappings
11. Equivalence Relations and Partititions
II. THE REAL NUMBERS
13. The Real Numbers
14. The Natural Numbers
15. The Integers
16. The Rational Numbers
17. The Complex Numbers
2. The Theory of Groups
1. The Group Concept
2. Some Simple Consequences of the Definition of Group
3. Powers of Elements in a Group
4. Order of a Group; Order of a Group Element
5. Cyclic Groups
6. The Symmetric Groups
7. Cycles; Decomposition of Permutations into Disjoint Cycles
8. Full Transformation Groups
9. Restrictions of Binary Operations
11. A Discussion of Subgroups
12. The Alternating Group
13. The Congruence of Integers
14. The Modular Arithmetics
15. Equivalence Relations and Subgroups
16. Index of a Subgroup
17. "Stable Relations, Normal Subgroups, Quotient Groups"
3. Group Isomorphism and Homomorphism
2. "Group Isomorphism; Examples, Definitions and Simplest Properties"
3. The Isomorphism Theorem for the Symmetric Groups
4. The Theorem of Cayley
5. Group Homomorphisms
6. A Relation Between Epimorphisms and Isomorphisms
7. Endomorphisms of a Group
4. The Theory of Rings
2. Definition of Ring
3. Some Properties of Rings
4. "The Modular Arithmetics, Again"
5. Integral Domains
8. Ring Homomorphisms
10. Residue Class Rings
11. Some Basic Homomorphism Theorems
12. Principle Ideal and Unique Factorization Domains
13. Prime and Maximal Ideals
14. The Quotient Field of an Integral Domain
5. Polynomial Rings
1. Introduction; The Concept of Polynomial Ring
3. Existence of Indeterminates
4. Polynomial Domains Over a Field
5. Unique Factorization in Polynomial Domains
6. Polynomial Rings in Two Indeterminates
7. Polynomial Functions and Polynomials
8. Some Characterizations of Intermediates
9. Substitution Homomorphisms
10. Roots of Polynomials