9781577664437

Abstract Algebra

by ;
  • ISBN13:

    9781577664437

  • ISBN10:

    1577664434

  • Edition: 3rd
  • Format: Hardcover
  • Copyright: 1/5/2006
  • Publisher: Waveland Pr Inc

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Table of Contents

Preface vii
Preface to the Second Edition ix
To the Student xiii
Writing Proofs xvi
Historical Background xxi
Integers
1(46)
Divisors
3(12)
Primes
15(9)
Congruences
24(11)
Integers Modulo n
35(12)
Notes
46(1)
Functions
47(40)
Functions
49(13)
Equivalence Relations
62(9)
Permutations
71(16)
Notes
86(1)
Groups
87(92)
Definition of a Group
88(14)
Subgroups
102(13)
Constructing Examples
115(9)
Isomorphisms
124(11)
Cyclic Groups
135(7)
Permutation Groups
142(10)
Homomorphisms
152(12)
Cosets, Normal Subgroups, and Factor Groups
164(15)
Notes
177(2)
Polynomials
179(44)
Fields; Roots of Polynomials
179(13)
Factors
192(11)
Existence of Roots
203(8)
Polynomials over Z, Q, R, and C
211(12)
Notes
222(1)
Commutative Rings
223(46)
Commutative Rings; Integral Domains
224(12)
Ring Homomorphisms
236(15)
Ideals and Factor Rings
251(11)
Quotient Fields
262(7)
Notes
268(1)
Fields
269(46)
Algebraic Elements
270(6)
Finite and Algebraic Extensions
276(7)
Geometric Constructions
283(6)
Splitting Fields
289(6)
Finite Fields
295(6)
Irreducible Polynomials over Finite Fields
301(6)
Quadratic Reciprocity
307(8)
Notes
314(1)
Structure of Groups
315(50)
Isomorphism Theorems; Automorphisms
316(7)
Conjugacy
323(7)
Groups Acting on Sets
330(8)
The Sylow Theorems
338(4)
Finite Abelian Groups
342(8)
Solvable Groups
350(7)
Simple Groups
357(8)
Galois Theory
365(42)
The Galois Group of a Polynomial
366(6)
Multiplicity of Roots
372(4)
The Fundamental Theorem of Galois Theory
376(10)
Solvability by Radicals
386(6)
Cyclotomic Polynomials
392(5)
Computing Galois Groups
397(10)
Unique Factorization
407(26)
Principal Ideal Domains
408(7)
Unique Factorization Domains
415(6)
Some Diophantine Equations
421(12)
APPENDIX
433(28)
Sets
433(3)
Construction of the Number Systems
436(3)
Basic Properties of the Integers
439(1)
Induction
440(4)
Complex Numbers
444(6)
Solution of Cubic and Quartic Equations
450(8)
Dimension of a Vector Space
458(3)
Bibliography 461(2)
Selected Answers 463(8)
Index of Symbols 471(4)
Index 475

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