The emphasis in this book is on examples and exercises, and they provide much of the motivation for the material. The author have also tried to provide some historical comment and to examine the connections between modern algebra and other fields. (These comments and connections often appear in the exercises.) In particular he have tried to point out the connections with elementary mathematics as they occur. A special cross-reference index has also been included for this purpose.
The basic plan of the book is as follows: The first two chapters cover groups, homomorphisms, and rings. Set theory, functions, equivalence relations, the integers, and the complex numbers are developed when needed, rather than in a chapter. A trimester or semester course can cover the first two chapters, especially if the sections marked t are omitted. The rest of the book is devoted to various topics which might be covered in a second trimester or semester. The main options are:
1 Groups acting on sets
2 Integral domains, number theory
3 Vector spaces (Secs. 3.1 and 3.2), further group theory (Sec. 5.1), field theory
4 Vector spaces, Wedderburn theorems, group representations
5 Wedderburn theorems
Chapter 4, "Groups Acting on Sets," is somewhat novel. It provides a chance to do some significant applications, such as counting benzene-based molecules. It also provides a more unified approach to the Sylow theorems, the class equation, Cayley's theorem with respect to a subgroup, and permutation groups. The first two sections are very easy, and he hopes the student will read them.
Another novelty is the three concluding survey chapters, designed to introduce the reader to some more advanced topics. The approach to this material is again through examples and exercises, and here many of the proofs are omitted.
Thorough references to the literature are given, and he hopes many readers will find these chapters useful as a first introduction to the material.
Sections marked * in the first two chapters may be omitted or treated lightly by anyone already acquainted with their contents. Sections marked t are not mandatory for a reading of Chapters 1 and 2 or for a first course.
Many exercises, are included. I have tried to place the exercises strategically, hoping the pedagogical gain outweighs the loss of smoothness of exposition.
Answers are provided for some of the exercises, hints for some others;overall, the answer section is extensive. The answers and hints should only be used as a last resort, or to check an answer. Mathematics is learned through idea formation and, though this may take different forms, looking up answers in the back of the book is not one of these.
The pace of Chapters 1 and 2 is leisurely, and the pace of the remaining chapters moderate. he has attempted to write a long, easy book rather than a short, hard one.
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Title: Introduction to Abstract Algebra
Copyright
ยฉ 1975 by McGraw-Hili,
ISBN 0-07-056415-9
CONTENTS
PREFACE
Chapter 1: GROUPS
*1.1 SETS AND BINARY OPERATIONS
1.2 GROUPS, DEFINITIONS AND EXAMPLES
1.3 ELEMENTARY PROPERTIES OF GROUPS
1.4 SUBGROUPS AND CYCLIC GROUPS
*1.5 FUNCTIONS
1.6 2 x 2 MATRICES
1.7 PERMUTATIONS
*1.8 EQUIVALENCE RELATIONS
1.9 LAGRANGE'S THEOREM
1.10 ISOMORPHISMS
*1.11 EUCLID'S ALGORITHM AND THE LINEAR PROPERTY
1.12 CYCLIC GROUPS AND DIRECT PRODUCTS
1.13 HOMOMORPHISMS AND NORMAL SUBGROUPS
*1.14 THE COMMUTATOR SUBGROUP AND A UNIVERSAL MAPPING PROPERTY
*1.15 ODDS AND ENDS
*l.16 SOME HISTORICAL REMARKS
Chapter 2: RINGS
2.1 DEFINITIONS AND EXAMPLES
*2.2 TWO IMPORTANT EXAMPLES
2.3 MATRIX RINGS
2.4 SUBRINGS, IDEALS, AND RING HOMOMORPHISMS
2.5 FACTORING OUT MAXIMAL IDEALS IN A COMMUTATIVE RING
2.6 POLYNOMIAL RINGS
Chapter 3: VECTOR SPACES
3.1 BASIC DEFINITIONS
3.2 BASES AND DIMENSION
3.3 LINEAR TRANSFORMATIONS
3.4 MATRICES
Chapter 4: GROUPS ACTING ON SETS
4.1 BASIC DEFINITIONS
4.2 FIXED POINTS OF p-GROUPS
4.3 THE BURNSIDE COUNTING THEOREM
4.4 SOME APPLICATIONS IN GROUP THEORY
Chapter 5: FURTHER RESULTS ON GROUPS
5.1 SOLVABLE GROUPS
5.2 FINITELY GENERATED ABELIAN GROUPS
Chapter 6: INTEGRAL DOMAINS
6.1 DEFINITIONS AND QUOTIENT FIELDS
6.2 EUCLIDEAN DOMAINS AND PRINCIPAL IDEAL DOMAINS
6.3 UNIQUE FACTORIZATION DOMAINS AND APPLICATIONS
6.4 ODDS AND ENDS
Chapter 7: NUMBER THEORY
7.1 BASIC RESULTS
7.2 QUADRATIC RESIDUES
7.3 THE TWO-SQUARE THEOREM OF FERMAT
Chapter 8: THE WEDDERBURN THEOREMS (A SURVEY)
Chapter 9: GROUP REPRESENTATIONS (A SURVEY)
9.1 REPRESENTATIONS
9.2 CHARACTERS
Chapter 10: A SURVEY OF GALOIS THEORY
10.1 SOME FIELD THEORY
10.2 GEOMETRIC CONSTRUCTIONS AND IMPOSSIBILITY THEOREMS
10.3 GALOIS THEORY (PRELIMINARIES)
10.4 GALOIS THE OR Y (FUNDAMENTAL THEOREM)
BIBLIOGRAPHY
SELECTED ANSWERS AND OCCASIONAL HINTS AND COMMENTS
CROSS-REFERENCES TO ELEMENTARY MATHEMATICS
NOTATION
SUBJECT INDEX