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

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
• 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
• 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
• Refining the Root Tower
• Solvable Galois Groups

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