Elements of Abstract Algebra

Foreword (v)......Page 4
Contents (vii)......Page 6
Introduction (ix)......Page 8
Ch. 1. Set Theory (1)......Page 17
The Notation and Terminology of Set Theory (2)......Page 18
Mappings (5)......Page 21
Equivalence Relations (9)......Page 25
Properties of the Natural Numbers (11)......Page 27
The Definition of Group Structure (17)......Page 33
Examples of Group Structure (22)......Page 38
Subgroups and Cosets (24)......Page 40
Conjugacy, Normal Subgroups, and Quotients Groups (32)......Page 48
The Sylow Theorems (39)......Page 55
Group Homomorphism and Isomorphism (45)......Page 61
Normal and Composition Series (53)......Page 69
The Symmetric Groups (57)......Page 73
Ch. 3. Field Theory (66)......Page 82
Definition and Examples of Field Structure (67)......Page 83
Vector Spaces, Bases, and Dimension (69)......Page 85
Extension Fields (73)......Page 89
Polynomials (75)......Page 91
Algebraic Extensions (88)......Page 104
Constructions with Straightedge and Compass (95)......Page 111
Ch. 4. Galois Theory (103)......Page 119
Automorphisms (104)......Page 120
Galois Extensions (108)......Page 124
Solvability of Equations by Radicals (130)......Page 146
Definition and Examples of Ring Structure (145)......Page 161
Ideals (151)......Page 167
Unique Factorization (160)......Page 176
Fields of Fractions (174)......Page 190
Dedekind Domains (179)......Page 195
Integral Extensions (186)......Page 202
Algebraic Integers (188)......Page 204
General References (197)......Page 213
Historical References (198)......Page 214
Index (199)......Page 215
Symbols for Special Sets (204)......Page 220