First Course in Abstract Algebra, A

Series
Addison-Wesley
Author
Joseph J. Rotman  
Publisher
Pearson
Cover
Softcover
Edition
3
Language
English
Total pages
640
Pub.-date
September 2005
ISBN13
9780131862678
ISBN
0131862677
Related Titles



Description

This text introduces students to the algebraic concepts of group and rings, providing a comprehensive discussion of theory as well as a significant number of applications for each.

Features

• Comprehensive coverage of abstract algebra - Includes discussions of the fundamental theorem of Galois theory; Jordan-Holder theorem; unitriangular groups; solvable groups; construction of free groups; von Dyck's theorem, and presentations of groups by generators and relations.

• Significant applications for both group and commutative ring theories, especially with Gr รถ bner bases -  Helps students see the immediate value of abstract algebra.

• Flexible presentation - May be used to present both ring and group theory in one semester, or for two-semester course in abstract algebra.

Number theory - Presents concepts such as induction, factorization into primes, binomial coefficients and DeMoivre's Theorem, so students can learn to write proofs in a familiar context.

• Section on Euclidean rings - Demonstrates that the quotient and remainder from the division algorithm in the Gaussian integers may not be unique. Also, Fermat's Two-Squares theorem is proved.

Sylow theorems - Discusses the existence of Sylow subgroups as well as conjugacy and the congruence condition on their number.

• Fundamental theorem of finite abelian groups - Covers the basis theorem as well as the uniqueness to isomorphism

• Extensive references and consistent numbering system for lemmas, theorems, propositions, corollaries, and examples - Clearly organized notations, hints, and appendices simplify student reference.

New to this Edition

• Rewritten for smoother exposition - Makes challenging material more accessible to students.

• Updated exercises - Features challenging new problems, with redesigned page and back references for easier access.

• Extensively revised Ch. 2 (groups) and Ch. 3 (commutative rings ) - Makes chapters independent of one another, giving instructors increased flexibility in course design.

• New coverage of codes -  Includes 28-page introduction to codes, including a proof that Reed-Solomon codes can be decoded.

• New section on canonical forms (Rational, Jordan, Smith) for matrices - Focuses on the definition and basic properties of exponentiation of complex matrices, and why such forms are valuable.

• New classification of frieze groups - Discusses why viewing the plane as complex numbers allows one to describe all isometries with very simple formulas.

• Expanded discussion of orthogonal Latin squares - Includes coverage of magic squares.

• Special Notation section - References common symbols and the page on which they are introduced.

Table of Contents

Chapter 1: Number Theory

Induction

Binomial Coefficients

Greatest Common Divisors

The Fundamental Theorem of Arithmetic

Congruences

Dates and Days

 

Chapter 2: Groups I

Some Set Theory

Permutations

Groups

Subgroups and Lagrange's Theorem

Homomorphisms

Quotient Groups

Group Actions

Counting with Groups

 

Chapter 3: Commutative Rings I

First Properties

Fields

Polynomials

Homomorphisms

Greatest Common Divisors

Unique Factorization

Irreducibility

Quotient Rings and Finite Fields

Officers, Magic, Fertilizer, and Horizons

 

Chapter 4: Linear Algebra

Vector Spaces

Euclidean Constructions

Linear Transformations

Determinants

Codes

Canonical Forms

 

Chapter 5: Fields

Classical Formulas

Insolvability of the General Quintic

Epilog

 

Chapter 6: Groups II

Finite Abelian Groups

The Sylow Theorems

Ornamental Symmetry

 

Chapter 7: Commutative Rings III

Prime Ideals and Maximal Ideals

Unique Factorization

Noetherian Rings

Varieties

Grobner Bases

 

Hints for Selected Exercises

Bibliography

Index


Instructor Resources