A First Course in Abstract Algebra

by Rotman, Joseph J.

• ISBN13: 9780131862678
• ISBN10: 0131862677
• Edition: 3rd
• Format: Paperback
• Copyright: 2005-09-28
• Publisher: Pearson

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 Edition	p. ix
Suggested Syllabi	p. xiii
To the Reader	p. xvii
Number Theory	p. 1
Induction	p. 1
Binomial Theorem and Complex Numbers	p. 18
Greatest Common Divisors	p. 37
The Fundamental Theorem of Arithmetic	p. 55
Congruences	p. 59
Dates and Days	p. 76
Groups I	p. 84
Some Set Theory	p. 84
Functions	p. 87
Equivalence Relations	p. 99
Permutations	p. 106
Groups	p. 125
Symmetry	p. 137
Subgroups and Lagrange's Theorem	p. 147
Homomorphisms	p. 159
Quotient Groups	p. 171
Group Actions	p. 192
Counting with Groups	p. 208
Commutative Rings I	p. 217
First Properties	p. 217
Fields	p. 230
Polynomials	p. 235
Homomorphisms	p. 243
From Numbers to Polynomials	p. 252
Euclidean Rings	p. 267
Unique Factorization	p. 275
Irreducibility	p. 281
Quotient Rings and Finite Fields	p. 290
A Mathematical Odyssey	p. 305
Latin Squares	p. 305
Magic Squares	p. 310
Design of Experiments	p. 314
Projective Planes	p. 316
Linear Algebra	p. 320
Vector Spaces	p. 320
Gaussian Elimination	p. 344
Euclidean Constructions	p. 354
Linear Transformations	p. 366
Eigenvalues	p. 383
Codes	p. 399
Block Codes	p. 399
Linear Codes	p. 406
Decoding	p. 423
Fields	p. 432
Classical Formulas	p. 432
Viete's Cubic Formula	p. 444
Insolvability of the General Quintic	p. 449
Formulas and Solvability by Radicals	p. 459
Quadratics	p. 460
Cubics	p. 461
Quartics	p. 461
Translation into Group Theory	p. 462
Epilog	p. 471
Groups II	p. 475
Finite Abelian Groups	p. 475
The Sylow Theorems	p. 489
Ornamental Symmetry	p. 501
Commutative Rings II	p. 518
Prime Ideals and Maximal Ideals	p. 518
Unique Factorization	p. 525
Noetherian Rings	p. 535
Varieties	p. 540
Generalized Divison Algorithm	p. 558
Monomial Orders	p. 559
Division Algorithm	p. 565
Grobner Bases	p. 570
Inequalities	p. 1
Pseudocodes	p. 3
Hints for Selected Exercises	p. 1
Bibliography	p. 1
Index	p. 1
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