9780201763904

A First Course in Abstract Algebra

by
  • ISBN13:

    9780201763904

  • ISBN10:

    0201763907

  • Edition: 7th
  • Format: Hardcover
  • Copyright: 11/6/2002
  • Publisher: Pearson

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Summary

Considered a classic by many,A First Course in Abstract Algebra, Seventh Edition is an in-depth introduction to abstract algebra. Focused on groups, rings and fields, this text gives students a firm foundation for more specialized work by emphasizing an understanding of the nature of algebraic structures. Sets and Relations; GROUPS AND SUBGROUPS; Introduction and Examples; Binary Operations; Isomorphic Binary Structures; Groups; Subgroups; Cyclic Groups; Generators and Cayley Digraphs; PERMUTATIONS, COSETS, AND DIRECT PRODUCTS; Groups of Permutations; Orbits, Cycles, and the Alternating Groups; Cosets and the Theorem of Lagrange; Direct Products and Finitely Generated Abelian Groups; Plane Isometries; HOMOMORPHISMS AND FACTOR GROUPS; Homomorphisms; Factor Groups; Factor-Group Computations and Simple Groups; Group Action on a Set; Applications of G-Sets to Counting; RINGS AND FIELDS; Rings and Fields; Integral Domains; Fermat's and Euler's Theorems; The Field of Quotients of an Integral Domain; Rings of Polynomials; Factorization of Polynomials over a Field; Noncommutative Examples; Ordered Rings and Fields; IDEALS AND FACTOR RINGS; Homomorphisms and Factor Rings; Prime and Maximal Ideas; Grouml;bner Bases for Ideals; EXTENSION FIELDS; Introduction to Extension Fields; Vector Spaces; Algebraic Extensions; Geometric Constructions; Finite Fields; ADVANCED GROUP THEORY; Isomorphism Theorems; Series of Groups; Sylow Theorems; Applications of the Sylow Theory; Free Abelian Groups; Free Groups; Group Presentations; GROUPS IN TOPOLOGY; Simplicial Complexes and Homology Groups; Computations of Homology Groups; More Homology Computations and Applications; Homological Algebra; Factorization; Unique Factorization Domains; Euclidean Domains; Gaussian Integers and Multiplicative Norms; AUTOMORPHISMS AND GALOIS THEORY; Automorphisms of Fields; The Isomorphism Extension Theorem; Splitting Fields; Separable Extensions; Totally Inseparable Extensions; Galois Theory; Illustrations of Galois Theory; Cyclotomic Extensions; Insolvability of the Quintic; Matrix Algebra For all readers interested in abstract algebra.

Table of Contents

Instructor's Preface vii
Student's Preface xi
Dependence Chart xiii
Sets and Relations
1(10)
I Groups and Subgroups 11(64)
Introduction and Examples
11(9)
Binary Operations
20(8)
Isomorphic Binary Structures
28(8)
Groups
36(13)
Subgroups
49(10)
Cyclic Groups
59(9)
Generating Sets and Cayley Digraphs
68(7)
II Permutations, Cosets, and Direct Products 75(50)
Groups of Permutations
75(12)
Orbits, Cycles, and the Alternating Groups
87(9)
Cosets and the Theorem of Lagrange
96(8)
Direct Products and Finitely Generated Abelian Groups
104(10)
Plane Isometries
114(11)
III Homomorphisms and Factor Groups 125(42)
Homomorphisms
125(10)
Factor Groups
135(9)
Factor-Group Computations and Simple Groups
144(10)
Group Action on a Set
154(7)
Applications of G-Sets to Counting
161(6)
IV Rings and Fields 167(70)
Rings and Fields
167(10)
Integral Domains
177(7)
Fermat's and Euler's Theorems
184(6)
The Field of Quotients of an Integral Domain
190(8)
Rings of Polynomials
198(11)
Factorization of Polynomials over a Field
209(11)
Noncommutative Examples
220(7)
Ordered Rings and Fields
227(10)
V Ideals and Factor Rings 237(28)
Homomorphisms and Factor Rings
237(8)
Prime and Maximal Ideals
245(9)
Grobner Bases for Ideals
254(11)
VI Extension Fields 265(42)
Introduction to Extension Fields
265(9)
Vector Spaces
274(9)
Algebraic Extensions
283(10)
Geometric Constructions
293(7)
Finite Fields
300(7)
VII Advanced Group Theory 307(48)
Isomorphism Theorems
307(4)
Series of Groups
311(10)
Sylow Theorems
321(6)
Applications of the Sylow Theory
327(6)
Free Abelian Groups
333(8)
Free Groups
341(5)
Group Presentations
346(9)
VIII Groups in Topology 355(34)
Simplicial Complexes and Homology Groups
355(8)
Computations of Homology Groups
363(8)
More Homology Computations and Applications
371(8)
Homological Algebra
379(10)
IX Factorization 389(26)
Unique Factorization Domains
389(12)
Euclidean Domains
401(6)
Gaussian Integers and Multiplicative Norms
407(8)
X Automorphisms and Galois Theory 415(62)
Automorphisms of Fields
415(9)
The Isomorphism Extension Theorem
424(7)
Splitting Fields
431(5)
Separable Extensions
436(8)
Totally Inseparable Extensions
444(4)
Galois Theory
448(9)
Illustrations of Galois Theory
457(7)
Cyclotomic Extensions
464(6)
Insolvability of the Quintic
470(7)
Appendix: Matrix Algebra 477(6)
Bibliography 483(4)
Notations 487(4)
Answers to Odd-Numbered Exercises Not Asking for Definitions or Proofs 491(22)
Index 513

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