**Download Free PDF | A First Course in Abstract Algebra Rings, Groups, and Fields 3rd Edition by Marlow Anderson and Todd Feil.**

## Contents of Course in Abstract Algebra

**Numbers Polynomials and Factoring**- 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
- Section I in a Nutshell
**II Rings Domains and Fields**- 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
- Section II in a Nutshell
**III Ring Homomorphisms and Ideals**- 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
- Section III in a Nutshell
**IV Groups**- 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 Groups
- 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
- Section IV in a Nutshell
**V Group Homomorphisms**- 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
- Section V in a Nutshell
**VI Topics from Group Theory**- 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
- Section VI in a Nutshell
**VII Unique Factorization**- 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
- A Comparison between Z and Z
- 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
- Section VII in a Nutshell
**VIII Constructibility Problems**- 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
- Section VIII in a Nutshell
**IX Vector Spaces and Field Extensions**- 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
- Section IX in a Nutshell
**X Galois Theory**- 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