A First Course in Abstract Algebra Rings, Groups, and Fields 3rd Edition by Marlow Anderson and Todd Feil

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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
  • 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

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All books on this website are published in good faith and for educational information purpose only. So, we ask you to report us any copyrighted material published in our website and we will remove it immediately.