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.
Author(s): Joseph J. Rotman
Edition: 3
Publisher: Pearson Prentice Hall
Year: 2005
Language: English
Pages: 616 [629]
City: Upper Saddle River
Tags: Rotman, Algebra, Abstract Algebra
Title Page
Contents
Preface
1 Number theory
1.1 Induction
1.2 Binomial Coefficients
1.3 Greatest Common Divisors
1.4 The Fundamental Theorem of Arithmetic
1.5 Congruences
1.6 Dates and Days
2 Groups I
2.1 Some Set Theory
Functions
Equivalence relations
2.2 Permutations
2.3 Groups
Symmetry
2.4 Subgroups and Lagrange's Theorem
2.5 Homomorphisms
2.6 Quotient Groups
2.7 Group Actions
2.8 Counting with Groups
3 Rings I
3.1 First properties
3.2 Fields
3.3 Polynomials
3.4 Homomorphisms
3.5 Greatest Common Divisors
Euclidean Rings
3.6 Unique Factorization
3.7 Irreducibility
3.8 Quotient Rings and Finite Fields
3.9 Officers, Magic, Fertilizer, and Horizons
Officers
Magic
Fertilizer
Horizons
4 Linear Algebra
4.1 Vector Spaces
Gaussian Elimination
4.2 Euclidean Constructions
4.3 Linear Transformations
3.4 Determinants
3.5 Codes
Block Codes
Linear Codes
5 Fields
4.1 Classical Formulas
Viète's Cubic Formula
4.2 Insolvability of the General Quintic
Formulas and Solvability by Radicals
Translation into Group Theory
5.3 Epilog
6 Groups II
6.1 Finite Abelian Groups
6.2 The Sylow Theorems
6.3 Ornamental Symmetry
7 Commutative Rings II
7.1 Prime Ideals and Maximal Ideals
7.2 Unique Factorization
7.3 Noetherian Rings
7.4 Varieties
7.5 Gröbner Bases
Monomial Orders
Generalized Division Algorithm
Gröbner Bases
Appendix A
Appendix B
Hints for Selected Exercises
Bibliography
Index
