A First Course in Abstract Algebra Rings, Groups, and Fields, Third Edition

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

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 2014

Praise 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