A First Course in Abstract Algebra

Front Cover
Title Page
Copyright Page
Contents
Instructor’s Preface
Dependence Chart
Student’s Preface
0 Sets and Relations
I Groups and Subgroups
	1 Binary Operations
	2 Groups
	3 Abelian Examples
	4 Nonabelian Examples
	5 Subgroups
	6 Cyclic Groups
	7 Generating Sets and Cayley Digraphs
II Structure of Groups
	8 Groups of Permutations
	9 Finitely Generated Abelian Groups
	10 Cosets and the Theorem of Lagrange
	11 Plane Isometries
III Homomorphisms and Factor Groups
	12 Factor Groups
	13 Factor-Group Computations and Simple Groups
	14 Group Action on a Set
	15 Applications of G-Sets to Counting
IV Advanced Group Theory
	16 Isomorphism Theorems
	17 Sylow Theorems
	18 Series of Groups
	19 Free Abelian Groups
	20 Free Groups
	21 Group Presentations
V Rings and Fields
	22 Rings and Fields
	23 Integral Domains
	24 Fermat’s and Euler’s Theorems
	25 Encryption
VI Constructing Rings and Fields
	26 the Field of Quotients of an Integral Domain
	27 Rings of Polynomials
	28 Factorization of Polynomials over a Field
	29 Algebraic Coding Theory
	30 Homomorphisms and Factor Rings
	31 Prime and Maximal Ideals
	32 Noncommutative Examples
VII Commutative Algebra
	33 Vector Spaces
	34 Unique Factorization Domains
	35 Euclidean Domains
	36 Number Theory
	37 Algebraic Geometry
	38 Gröbner Bases for Ideals
VIII Extension Fields
	39 Introduction to Extension Fields
	40 Algebraic Extensions
	41 Geometric Constructions
	42 Finite Fields
IX Galois Theory
	43 Introduction to Galois Theory
	44 Splitting Fields
	45 Separable Extensions
	46 Galois Theory
	47 Illustrations of Galois Theory
	48 Cyclotomic Extensions
	49 Insolvability of the Quintic
Appendix: Matrix Algebra
Bibliography
Notations
Answers to Odd-numbered Exercises Not Asking for Definitions or Proofs
Index