A First Course in Abstract Algebra Rings, Groups, and Fields Second Edition by Marlow Anderson and Todd Feil is a comprehensive textbook for undergraduate students who are studying abstract algebra for the first time. This book covers the fundamental concepts of rings, groups, and fields 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 book is written in a clear and concise manner and is rich in examples and exercises to help students understand the material.

The second edition of A First Course in Abstract Algebra Rings, Groups, and Fields has been updated and revised to reflect the latest advancements in the field. The book covers all the essential topics, including ring theory, group theory, and field theory. It also includes chapters on Galois theory and modular arithmetic, which are important topics in abstract algebra.

**TABLE OF CONTENTS**

**Chapter 1 The Natural Numbers**

- Operations on the Natural Numbers
- Well Ordering and Mathematical Induction
- The Fibonacci Sequence
- Well Ordering Implies Mathematical Induction
- The Axiomatic Method

**Chapter 2 The Integers**

- The Division Theorem
- The Greatest Common Divisor
- The GCD Identity
- The Fundamental Theorem of Arithmetic
- A Geometric Interpretation

**Chapter 3 Modular Arithmetic**

- Residue Classes
- Arithmetic on the Residue Classes
- Properties of Modular Arithmetic

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**Chapter 4 Polynomials with Rational Coefficients**

- Polynomials
- The Algebra of Polynomials
- The Analogy between Z and Q[x]
- Factors of Polynomial
- Linear Factors
- Greatest Common Divisors

**Chapter 5 Factorization of Polynomials**

- Factoring Polynomials
- Unique Factorization
- Polynomials with integer Coefficient

**Chapter 6 Rings**

- Binary Operations
- Rings
- Arithmetic in a Ring
- Notational Conventions
- The set of Integers is a Ring

**Chapter 7 Subrings and Unity**

- Subrings
- The Multiplicative Identity

**Chapter 8 Integral Domains and Fields**

- Zero Divisors
- Units
- Fields
- The Field of Complex Numbers
- Finite Fields

**Chapter 9 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

**Chapter 10 Associates and Irreducibles**

- Associates
- Irreducibles
- Quadratic Extensions of the Integers
- Units in Quadratic Extensions
- Irreducibles in Quadratic Extensions

**Chapter 11 Factorization and Ideals**

- Factorization for Quadratic Extensions
- How might Factorization Fails?
- Ideals
- Principal Ideals

**Chapter 12 Principal Ideal Domains**

- Ideals that is not Principal
- Principal Ideal Domain

**Chapter 13 Primes and Unique Factorization**

- Primes
- UFDs
- Expressing Properties of Elements in Terms of Ideals
- All PIDs and UFDs

**Chapter 14 Polynomials with Integer Co-efficient**

- The Proof that Q[x] is a UFD
- Factoring Integers out of Polynomials
- Irreducibles in Z[x] are Prime

**Chapter 15 Euclidean Domains**

- Euclidean Domains
- The Gaussian Integers
- Euclidean Domains are PIDs
- Some PIDs are Euclidean

**Chapter 16 Ring Homomorphisms**

- Homomorphisms
- One-to-One and Onto Functions
- Properties Preserved by Homomorphisms
- More Examples
- Making Homomorphisms Onto

**Chapter 17 The Kernel**

- Ideals
- The Kernel
- The Kernel is an Ideal
- All pre-images can be obtained from the Kernel
- What is the Kernel Trivial

**Chapter 18 Rings and Cosets**

- The Ring of Cosets
- The Natural Homomorphism

**Chapter 19 The Isomorphism Theorem for Rings**

- Isomorphism
- The Fundamental Isomorphism Theorem

**Chapter 20 Maximal and Prime Ideals**

- Maximal Ideals
- Prime Ideals

**Chapter 21 The Chinese Remainder Theorem**

- Direct Products of Domains
- Chinese Remainder Theorem

**Chapter 22 Symmetries of Figures in the Plane**

- Symmetries of the Equilateral Triangle
- Permutation Notation
- Matrix Notation
- Symmetries of the Square

**Chapter 23 Symmetries of Figures in Space**

- Symmetries of the Regular Tetrahedron
- Symmetries of the Cube

**Chapter 24 Abstract Groups**

- Definition of Group
- Examples of Groups
- Multiplicative Groups

**Chapter 25 Subgroups**

- Arithmetic in an Abstract Group
- Notation
- Subgroups
- Characterization of Subgroups

**Chapter 26 Cyclic Groups **

- The Order of an Element
- Rule of Exponents
- Cyclic Subgroups
- Cyclic Groups

**Chapter 27 Group Homomorphisms**

- Homomorphisms
- Examples
- Direct Products

**Chapter 28 Group Isomorphism**

- Structure Preserved by Homomorphisms
- Uniqueness of Cyclic Groups
- Symmetry Groups
- Characterizing Direct Products

**Chapter 29 Permutations and Cayley’s Theorem**

- Permutations
- The Symmetric Groups
- Cayley’s Theorem

**Chapter 30 More about Permutations**

- Cycles
- Cycles Factorization of Permutations
- Orders of Permutations

**Chapter 31 Cosets and Lagrange’s Theorem**

- Cosets
- Langranges Theorem
- Applications of Langranges Theorem

**Chapter 32 Groups of Cosets**

- Left Cosets
- Normal Subgroups
- Examples of Groups of Cosets

**Chapter 33 The Isomorphism Theorem for Groups**

- The Kernel
- Cosets of the Kernel
- The Fundamental Theorem

**Chapter 34 The Altering Groups**

- Transpositions
- The Parity of a Permutation
- The Altering Groups
- The Altering Subgroups is Normal
- Simple Groups

**Chapter 35 Fundamental Theorem for Finite Abelian Groups**

- The Fundamental Theorem
- p-groups

**Chapter 36 Solvable Groups**

- Solvability
- New Solvable Groups from Old

**Chapter 37 Constructions with Compass and Straightedge**

- Construction Problems
- Constructible Lengths and Numbers

**Chapter 38 Constructability and Quadratic Field Extensions**

- Quadratic Field Extensions
- Sequences of Quadratic Field Extensions
- The Rational Plane
- Planes of Constructible Numbers
- The Constructible Number Theory

**Chapter 39 The Impossibility of Certain Constructions**

- Doubling the Cube
- Trisecting the Angle
- Squaring the Circle

**Chapter 40 Vector Spaces I**

- Vectors
- Vector Spaces

**Chapter 41 Vector Spaces II**

- Spanning Sets
- A Basis for a Vector Space
- Finding a Basis
- Dimension of a Vector Space

**Chapter 42 Field Extensions and Kronecker’s Theorem**

- Field Extensions
- Kronecker’s Theorem
- The Characteristics of a Field

**Chapter 43 Algebraic Field Extensions**

- The Minimal Polynomial for an Element
- Simple Extensions
- Simple Transcendental Extensions
- Dimension of Simple Algebraic Extensions

**Chapter 44 Finite Extensions and Constructibility Revisited**

- Finite Extensions
- Constructability Problems

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One of the standout features of this book is its focus on examples and exercises. The authors provide numerous examples and exercises that 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.

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 Second 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 Second 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.