Librería Portfolio

Like its popular predecessors, A First Course in Abstract Algebra: Rings, Groups, and Fields, Third Edition develops ring theory first by drawing on students' familiarity with integers and polynomials. This unique approach motivates students in the study of abstract algebra and helps them understand the power of abstraction. The authors introduce groups later on using examples of symmetries of figures in the plane and space as well as permutations.

New to the Third Edition

Makes it easier to teach unique factorization as an optional topic
Reorganizes the core material on rings, integral domains, and fields
Includes a more detailed treatment of permutations
Introduces more topics in group theory, including new chapters on Sylow theorems
Provides many new exercises on Galois theory
The text includes straightforward exercises within each chapter for students to quickly verify facts, warm-up exercises following the chapter that test fundamental comprehension, and regular exercises concluding the chapter that consist of computational and supply-the-proof problems. Historical remarks discuss the history of algebra to underscore certain pedagogical points. Each section also provides a synopsis that presents important definitions and theorems, allowing students to verify the major topics from the section.



Numbers, Polynomials, and Factoring

The Natural Numbers

The Integers

Modular Arithmetic

Polynomials with Rational Coefficients

Factorization of Polynomials

Section I in a Nutshell

Rings, Domains, and Fields

Rings

Subrings and Unity

Integral Domains and Fields

Ideals

Polynomials over a Field

Section II in a Nutshell

Ring Homomorphisms and Ideals

Ring Homomorphisms

The Kernel

Rings of Cosets

The Isomorphism Theorem for Rings

Maximal and Prime Ideals

The Chinese Remainder Theorem

Section III in a Nutshell

Groups

Symmetries of Geometric Figures

Permutations

Abstract Groups

Subgroups

Cyclic Groups

Section IV in a Nutshell

Group Homomorphisms

Group Homomorphisms

Structure and Representation

Cosets and Lagrange´s Theorem

Groups of Cosets

The Isomorphism Theorem for Groups

Section V in a Nutshell

Topics from Group Theory

The Alternating Groups

Sylow Theory: The Preliminaries

Sylow Theory: The Theorems

Solvable Groups

Section VI in a Nutshell

Unique Factorization

Quadratic Extensions of the Integers

Factorization

Unique Factorization

Polynomials with Integer Coefficients

Euclidean Domains

Section VII in a Nutshell

Constructibility Problems

Constructions with Compass and Straightedge

Constructibility and Quadratic Field Extensions

The Impossibility of Certain Constructions

Section VIII in a Nutshell

Vector Spaces and Field Extensions

Vector Spaces I

Vector Spaces II

Field Extensions and Kronecker´s Theorem

Algebraic Field Extensions

Finite Extensions and Constructibility Revisited

Section IX in a Nutshell

Galois Theory

The Splitting Field

Finite Fields

Galois Groups

The Fundamental Theorem of Galois Theory

Solving Polynomials by Radicals

Section X in a Nutshell

Hints and Solutions

Guide to Notation

Index