9780131862678

A First Course in Abstract Algebra

by
  • ISBN13:

    9780131862678

  • ISBN10:

    0131862677

  • Edition: 3rd
  • Format: Paperback
  • Copyright: 9/28/2005
  • Publisher: Pearson

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Summary

This text introduces readers to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each. Number Theory:Induction; Binomial Coefficients; Greatest Common Divisors; The Fundamental Theorem of Arithmetic Congruences; Dates and Days.Groups I:Some Set Theory; Permutations; Groups; Subgroups and Lagrange's Theorem; Homomorphisms; Quotient Groups; Group Actions; Counting with Groups.Commutative Rings I:First Properties; Fields; Polynomials; Homomorphisms; Greatest Common Divisors; Unique Factorization; Irreducibility; Quotient Rings and Finite Fields; Officers, Magic, Fertilizer, and Horizons.Linear Algebra:Vector Spaces; Euclidean Constructions; Linear Transformations; Determinants; Codes; Canonical Forms.Fields:Classical Formulas; Insolvability of the General Quintic; Epilog.Groups II:Finite Abelian Groups; The Sylow Theorems; Ornamental Symmetry.Commutative Rings III:Prime Ideals and Maximal Ideals; Unique Factorization; Noetherian Rings; Varieties; Grobner Bases. For all readers interested in abstract algebra.

Table of Contents

Preface to the Third Editionp. ix
Suggested Syllabip. xiii
To the Readerp. xvii
Number Theoryp. 1
Inductionp. 1
Binomial Theorem and Complex Numbersp. 18
Greatest Common Divisorsp. 37
The Fundamental Theorem of Arithmeticp. 55
Congruencesp. 59
Dates and Daysp. 76
Groups Ip. 84
Some Set Theoryp. 84
Functionsp. 87
Equivalence Relationsp. 99
Permutationsp. 106
Groupsp. 125
Symmetryp. 137
Subgroups and Lagrange's Theoremp. 147
Homomorphismsp. 159
Quotient Groupsp. 171
Group Actionsp. 192
Counting with Groupsp. 208
Commutative Rings Ip. 217
First Propertiesp. 217
Fieldsp. 230
Polynomialsp. 235
Homomorphismsp. 243
From Numbers to Polynomialsp. 252
Euclidean Ringsp. 267
Unique Factorizationp. 275
Irreducibilityp. 281
Quotient Rings and Finite Fieldsp. 290
A Mathematical Odysseyp. 305
Latin Squaresp. 305
Magic Squaresp. 310
Design of Experimentsp. 314
Projective Planesp. 316
Linear Algebrap. 320
Vector Spacesp. 320
Gaussian Eliminationp. 344
Euclidean Constructionsp. 354
Linear Transformationsp. 366
Eigenvaluesp. 383
Codesp. 399
Block Codesp. 399
Linear Codesp. 406
Decodingp. 423
Fieldsp. 432
Classical Formulasp. 432
Viete's Cubic Formulap. 444
Insolvability of the General Quinticp. 449
Formulas and Solvability by Radicalsp. 459
Quadraticsp. 460
Cubicsp. 461
Quarticsp. 461
Translation into Group Theoryp. 462
Epilogp. 471
Groups IIp. 475
Finite Abelian Groupsp. 475
The Sylow Theoremsp. 489
Ornamental Symmetryp. 501
Commutative Rings IIp. 518
Prime Ideals and Maximal Idealsp. 518
Unique Factorizationp. 525
Noetherian Ringsp. 535
Varietiesp. 540
Generalized Divison Algorithmp. 558
Monomial Ordersp. 559
Division Algorithmp. 565
Grobner Basesp. 570
Inequalitiesp. 1
Pseudocodesp. 3
Hints for Selected Exercisesp. 1
Bibliographyp. 1
Indexp. 1
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