3rd Edition
A First Course in Abstract Algebra Rings, Groups, and Fields, Third Edition
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
Biography
Marlow Anderson, Todd Feil
"I am a fan of the rings-first approach to algebra, agreeing with the authors that students’ familiarity with the integers and with polynomials renders rings more intuitive and accessible than groups. But this book has many other virtues besides presenting the material in this order. For example, each section is preceded and followed by short sections that try to put the material into a broader context. … This is definitely a book worth considering for textbook adoption."
—MAA Reviews, November 2014Praise for the Second Edition:
"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
