A First Course in Abstract Algebra Rings, Groups, and Fields Third Edition by Marlow Anderson and Todd Feil is a comprehensive textbook that provides undergraduate students with an introduction to the fundamental concepts of abstract algebra. This book covers the essential topics of ring theory, group theory, and field theory, and provides students with the necessary tools to explore abstract algebra further.

The authors, Marlow Anderson and Todd Feil, are both experienced mathematicians and educators. They have written this book with the aim of making abstract algebra accessible to students with limited mathematical backgrounds. The third edition of the book has been updated and revised to reflect the latest advancements in the field.

One of the standout features of A First Course in Abstract Algebra Rings, Groups, and Fields Third Edition is its focus on examples and exercises. The authors provide numerous examples and exercises to help students develop their understanding of the material. The exercises are designed to be challenging, but not too difficult, and they help students reinforce the concepts they have learned.

**TABLE OF CONTENTS**

**The Natural Numbers**

Operations on the Natural Numbers

Well Ordering and Mathematical Induction

The Fibonacci Sequence

Well Ordering Implies Mathematical Induction

The Axiomatic Method

**The Integers**

The Division Theorem

The Greatest Common Divisor

The GCD Identity

The Fundamental Theorem of Arithmetic

A Geometric Interpretation

**Modular Arithmetic**

Residue Classes

Arithmetic on the Residue Classes

Properties of Modular Arithmetic

.

**Polynomials with Rational Coefficients**

Polynomials

The Algebra of Polynomials

The Analogy between Z and Q[x]

Factors of a Polynomial

Linear Factors

Greatest Common Divisors

**Factorization of Polynomials**

Factoring Polynomials

Unique Factorization

Polynomials with Integer Coefficients

**Rings**

Binary Operations

Rings

Arithmetic in a Ring

Notational Conventions

The Set of Integers Is a Ring

**Subrings and Unity**

Subrings

The Multiplicative Identity

Surjective, Injective, and Bijective Functions

Ring Isomorphisms

**Integral Domains and Fields**

Zero Divisors

Units

Associates

Fields

The Field of Complex Numbers

Finite Fields

**Ideals**

Principal Ideals

Ideals

Ideals That Are Not Principal

All Ideals in Z Are Principal

**Polynomials over a Field**

Polynomials with Coefficients from an Arbitrary Field

Polynomials with Complex Coefficients

Irreducibles in R[x]

Extraction of Square Roots in C

**Ring Homomorphisms**

Homomorphisms

Properties Preserved by Homomorphisms

More Examples

Making a Homomorphism Surjective

**The Kernel**

The Kernel

The Kernel Is an Ideal

All Pre-images Can Be Obtained from the Kernel

When Is the Kernel Trivial?

A Summary and Example

**Rings of Cosets**

The Ring of Cosets

The Natural Homomorphism

**The Isomorphism Theorem for Rings**

An Illustrative Example

The Fundamental Isomorphism Theorem

Examples

**Maximal and Prime Ideals**

Irreducibles

Maximal Ideals

Prime Ideals

An Extended Example

Finite Products of Domains

**The Chinese Remainder Theorem**

Some Examples

Chinese Remainder Theorem

A General Chinese Remainder Theorem

**Symmetries of Geometric Figures**

Symmetries of the Equilateral Triangle

Permutation Notation

Matrix Notation

Symmetries of the Square

Symmetries of Figures in Space

Symmetries of the Regular Tetrahedron

**Permutations**

Permutations

The Symmetric Group

Cycles

Cycle Factorization of Permutations

**Abstract Groups**

Definition of Group

Examples of Groups

Multiplicative Groups

**Subgroups**

Arithmetic in an Abstract Group

Notation

Subgroups

Characterization of Subgroups

Group Isomorphisms

**Cyclic Groups**

The Order of an Element

Rule of Exponents

Cyclic Subgroups

Cyclic Groups

**Group Homomorphisms**

Homomorphisms

Examples

Structure Preserved by Homomorphisms

Direct Products

**Structure and Representation**

Characterizing Direct Products

Cayley’s Theorem

**Cosets and Lagrange’s Theorem**

Cosets

Lagrange’s Theorem

Applications of Lagrange’s Theorem

**Groups of Cosets**

Left Cosets

Normal Subgroups

Examples of Groups of Cosets

**The Isomorphism Theorem for Groups**

The Kernel

Cosets of the Kernel

The Fundamental Theorem

**The Alternating Groups**

Transpositions

The Parity of a Permutation

The Alternating Groups

The Alternating Subgroup Is Normal

Simple Groups

**Sylow Theory: The Preliminaries**

p-groups

Groups Acting on Sets

**Sylow Theory: The Theorems**

The Sylow Theorems

Applications of the Sylow Theorems

The Fundamental Theorem for Finite Abelian Groups

**Solvable Groups**

Solvability

New Solvable Groups from Old

**Quadratic Extensions of the Integers**

Quadratic Extensions of the Integers

Units in Quadratic Extensions

Irreducibles in Quadratic Extensions

Factorization for Quadratic Extensions

**Factorization**

How Might Factorization Fail?

PIDs Have Unique Factorization

Primes

**Unique Factorization**

UFDs

All PIDs Are UFDs

**Polynomials with Integer Coefficients**

The Proof That Q[x] Is a UFD

Factoring Integers out of Polynomials

The Content of a Polynomial

Irreducibles in Z[x] Are Prime

**Euclidean Domains**

Euclidean Domains

The Gaussian Integers

Euclidean Domains Are PIDs

Some PIDs Are Not Euclidean

**Constructions with Compass and Straightedge**

Construction Problems

Constructible Lengths and Numbers

**Constructibility and Quadratic Field Extensions**

Quadratic Field Extensions

Sequences of Quadratic Field Extensions

The Rational Plane

Planes of Constructible Numbers

The Constructible Number Theorem

**The Impossibility of Certain Constructions **

Doubling the Cube

Trisecting the Angle

Squaring the Circle

**Vector Spaces I**

Vectors

Vector Spaces

**Vector Spaces II**

Spanning Sets

A Basis for a Vector Space

Finding a Basis

Dimension of a Vector Space

**Field Extensions and Kronecker’s Theorem**

Field Extensions

Kronecker’s Theorem

The Characteristic of a Field

**Algebraic Field Extensions**

The Minimal Polynomial for an Element

Simple Extensions

Simple Transcendental Extensions

Dimension of Algebraic Simple Extensions

**Finite Extensions and Constructibility Revisited**

Finite Extensions

Constructibility Problems

**The Splitting Field**

The Splitting Field

Fields with Characteristic Zero

**Finite Fields **

Existence and Uniqueness

Examples

**Galois Groups**

The Galois Group

Galois Groups of Splitting Fields

**The Fundamental Theorem of Galois Theory**

Subgroups and Subfields

Symmetric Polynomials

The Fixed Field and Normal Extensions

The Fundamental Theorem

Examples

**Solving Polynomials by Radicals**

Field Extensions by Radicals

Refining the Root Tower

Solvable Galois Groups

**Download Now Book in PDF**

The book is suitable for students who are studying abstract algebra for the first time. It assumes that students have a basic understanding of linear algebra and calculus, but no prior knowledge of abstract algebra is required. The authors have provided detailed explanations of the concepts, and have made the material accessible to students with limited mathematical backgrounds.

A First Course in Abstract Algebra Rings, Groups, and Fields Third Edition is an excellent resource for students who are studying abstract algebra. It provides a comprehensive introduction to the subject and is written in a clear and concise manner. The book is available in PDF format, which makes it easy for students to download and access.

In conclusion, A First Course in Abstract Algebra Rings, Groups, and Fields Third Edition by Marlow Anderson and Todd Feil is a must-have textbook for students who are studying abstract algebra. With its clear explanations, numerous examples, and challenging exercises, this book provides students with the foundation they need to succeed in their studies. The PDF format makes it easy for students to access and study the material, and the book is an excellent resource for anyone looking to learn about abstract algebra.