A First Course in Abstract Algebra: Rings, Groups and Fields, Second Edition
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Summary
Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there is more natural-and ultimately more effective.
Authors Anderson and Feil developed A First Course in Abstract Algebra: Rings, Groups and Fields based upon that conviction. The text begins with ring theory, building upon students' familiarity with integers and polynomials. Later, when students have become more experienced, it introduces groups. The last section of the book develops Galois Theory with the goal of showing the impossibility of solving the quintic with radicals.
Each section of the book ends with a "Section in a Nutshell" synopsis of important definitions and theorems. Each chapter includes "Quick Exercises" that reinforce the topic addressed and are designed to be worked as the text is read. Problem sets at the end of each chapter begin with "Warm-Up Exercises" that test fundamental comprehension, followed by regular exercises, both computational and "supply the proof" problems. A Hints and Answers section is provided at the end of the book.
As stated in the title, this book is designed for a first course--either one or two semesters in abstract algebra. It requires only a typical calculus sequence as a prerequisite and does not assume any familiarity with linear algebra or complex numbers.
Table of Contents
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
Polynomials over a Field
Section II in a Nutshell
UNIQUE FACTORIZATION
Associates and Irreducibles
Factorization and Ideals
Principal Ideal Domains
Primes and Unique Factorization
Polynomials with Integer Coefficients
Euclidean Domains
Section III 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 IV in a Nutshell
GROUPS
Symmetries of Figures in the Plane
Symmetries of Figures in Space
Abstract Groups
Subgroups
Cyclic Groups
Section V in a Nutshell
GROUP HOMOMORPHISMS AND PERMUTATIONS
Group Homomorphisms
Group Isomorphisms
Permutations and Cayley's Theorem
More About Permutations
Cosets and Lagrange's Theorem
Groups of Cosets
The Isomorphism Theorem for Groups
The Alternating Groups
Fundamental Theorem for Finite Abelian Groups
Solvable Groups
Section VI in a Nutshell
CONSTRUCTIBILITY PROBLEMS
Constructions with Compass and Straightedge
Constructibility and Quadratic Field Extensions
The Impossibility of Certain Constructions
Section VII 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 VIII in a Nutshell
GALOIS THEORY
The Splitting Field
Finite Fields
Galois Groups
The Fundamental Theorem of Galois Theory
Solving Polynomials by Radicals
Section IX in a Nutshell
Hints and Solutions
Guide to Notation
Index
Editorial Reviews
"I was quickly won over by the book … . The book is very complete, containing more than enough material for a two semester course in undergraduate abstract algebra … . Even though there was a great deal of material presented, I found the book to be very well organized. … There are a lot of things that I like about this book. … [It is] well written and will help students to see the big picture. … All in all it seems that a lot of thought went into this book, resulting in a comprehensive, well-written, readable book for undergraduates first learning abstract algebra."
- MAA Online
"A remarkable feature of the book is that it starts first with the concept of a ring, while groups are introduced later. The reason of that is that students are usually more familiar with various number domains rather than the mappings and matrices. There is a huge number of examples in the book….The book contains a lot of nice exercises of various degrees of difficulty so that it can also be used as a practice book."
-EMS Newsletter, March 2006
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