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.
