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



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A First Course in Abstract Algebra: Rings, Groups and Fields, (3rd Edition) written by Marlow Anderson, Colorado College, Colorado Springs, USA and Todd Feil, Denison University, Granville, Ohio, USA. Most abstract algebra texts begin with groups, then proceed to rings and fields. While groups are the logically simplest of the structures, the motivation for studying groups can be somewhat lost on students approaching abstract algebra for the first time. To engage and motivate them, starting with something students know and abstracting from there is more natural-and ultimately more effective. Authors Anderson and Feil developed A First Course in Abstract Algebra: Rings, Groups and Fields based upon that conviction. The text begins with ring theory, building upon students' familiarity with integers and polynomials. Later, when students have become more experienced, it introduces groups. The last 'of the book develops Galois Theory with the goal of showing the impossibility of solving the quintic with radicals. Each section of the book ends with a "Section in a Nutshell" synopsis of important definitions and theorems. Each chapter includes "Quick Exercises" that reinforce the topic addressed and are designed to be worked as the text is read. Problem sets at the end of each chapter begin with "Warm-Up Exercises" that test fundamental comprehension, followed by regular exercises, both computational and "supply the proof" problems. A Hints and Answers section is provided at the end of the book. As stated in the title, this book is designed for a first course--either one or two semesters in abstract algebra. It requires only a typical calculus sequence as a prerequisite and does not assume any familiarity with linear algebra or complex numbers.

A First Course in Abstract Algebra: Rings, Groups and Fields, (3rd Edition) by Marlow Anderson, Todd Feil written by Joseph A. Gallian cover the following topics.

  • I Numbers, Polynomials, and Factoring

  • 1. The Natural Numbers
    1.1 Operations on the Natural Numbers
    1.2 Well Ordering and Mathematical Induction
    1.3 The Fibonacci Sequence
    1.4 Well Ordering Implies Mathematical Induction
    1.5 The Axiomatic Method

  • 2. The Integers
    2.1 The Division Theorem
    2.2 The Greatest Common Divisor
    2.3 The GCD Identity
    2.4 The Fundamental Theorem of Arithmetic
    2.5 A Geometric Interpretation

  • 3. Modular Arithmetic
    3.1 Residue Classes
    3.2 Arithmetic on the Residue Classes
    3.3 Properties of Modular Arithmetic

  • 4. Polynomials with Rational Coecients
    4.1 Polynomials
    4.2 The Algebra of Polynomials
    4.3 The Analogy between Z and Q[x]
    4.4 Factors of a Polynomial
    4.5 Linear Factors
    4.6 Greatest Common Divisors

  • 5. Factorization of Polynomials
    5.1 Factoring Polynomials
    5.2 Unique Factorization
    5.3 Polynomials with Integer Coecients
    Section I in a Nutshell

  • II Rings, Domains, and Fields

  • 6. Rings
    6.1 Binary Operations
    6.2 Rings
    6.3 Arithmetic in a Ring
    6.4 Notational Conventions
    6.5 The Set of Integers Is a Ring

  • 7. Subrings and Unity
    7.1 Subrings
    7.2 The Multiplicative Identity
    7.3 Surjective, Injective, and Bijective Functions
    7.4 Ring Isomorphisms

  • 8. Integral Domains and Fields
    8.1 Zero Divisors
    8.2 Units
    8.3 Associates
    8.4 Fields
    8.5 The Field of Complex Numbers
    8.6 Finite Fields

  • 9. Ideals
    9.1 Principal Ideals
    9.2 Ideals
    9.3 Ideals That Are Not Principal
    9.4 All Ideals in Z Are Principal

  • 10. Polynomials over a Field
    10.1 Polynomials with Coecients from an Arbitrary Field
    10.2 Polynomials with Complex Coecients
    10.3 Irreducibles in R[x]
    10.4 Extraction of Square Roots in C
    Section II in a Nutshell

  • III Ring Homomorphisms and Ideals

  • 11. Ring Homomorphisms
    11.1 Homomorphisms
    11.2 Properties Preserved by Homomorphisms
    11.3 More Examples
    11.4 Making a Homomorphism Surjective

  • 12. The Kernel
    12.1 The Kernel
    12.2 The Kernel Is an Ideal
    12.3 All Pre-images Can Be Obtained from the Kernel
    12.4 When Is the Kernel Trivial?
    12.5 A Summary and Example

  • 13. Rings of Cosets
    13.1 The Ring of Cosets
    13.2 The Natural Homomorphism

  • 14. The Isomorphism Theorem for Rings
    14.1 An Illustrative Example
    14.2 The Fundamental Isomorphism Theorem
    14.3 Examples

  • 15. Maximal and Prime Ideals
    15.1 Irreducibles
    15.2 Maximal Ideals
    15.3 Prime Ideals
    15.4 An Extended Example
    15.5 Finite Products of Domains

  • 16. The Chinese Remainder Theorem
    16.1 Some Examples
    16.2 Chinese Remainder Theorem
    16.3 A General Chinese Remainder Theorem
    Section III in a Nutshell

  • V Groups

  • 17. Symmetries of Geometric Figures
    17.1 Symmetries of the Equilateral Triangle
    17.2 Permutation Notation
    17.3 Matrix Notation
    17.4 Symmetries of the Square
    17.5 Symmetries of Figures in Space
    17.6 Symmetries of the Regular Tetrahedron

  • 18. Permutations
    18.1 Permutations
    18.2 The Symmetric Groups
    18.3 Cycles
    18.4 Cycle Factorization of Permutations

  • 19. Abstract Groups
    19.1 De nition of Group
    19.2 Examples of Groups
    19.3 Multiplicative Groups

  • 20. Subgroups
    20.1 Arithmetic in an Abstract Group
    20.2 Notation
    20.3 Subgroups
    20.4 Characterization of Subgroups
    20.5 Group Isomorphisms

  • 21. Cyclic Groups
    21.1 The Order of an Element
    21.2 Rule of Exponents
    21.3 Cyclic Subgroups
    21.4 Cyclic Groups
    Section IV in a Nutshell

  • VI Group Homomorphisms

  • 22. Group Homomorphisms
    22.1 Homomorphisms
    22.2 Examples
    22.3 Structure Preserved by Homomorphisms
    22.4 Direct Products

  • 23. Structure and Representation
    23.1 Characterizing Direct Products
    23.2 Cayley's Theorem

  • 24. Cosets and Lagrange's Theorem
    24.1 Cosets
    24.2 Lagrange's Theorem
    24.3 Applications of Lagrange's Theorem

  • 25. Groups of Cosets
    25.1 Left Cosets
    25.2 Normal Subgroups
    25.3 Examples of Groups of Cosets

  • 26. The Isomorphism Theorem for Groups
    26.1 The Kernel
    26.2 Cosets of the Kernel
    26.3 The Fundamental Theorem
    Section V in a Nutshell

  • VI Topics from Group Theory

  • 27. The Alternating Groups
    27.1 Transpositions
    27.2 The Parity of a Permutation
    27.3 The Alternating Groups
    27.4 The Alternating Subgroup Is Normal
    27.5 Simple Groups

  • 28. Sylow Theory: The Preliminaries
    28.1 p-groups
    28.2 Groups Acting on Sets

  • 29. Sylow Theory: The Theorems
    29.1 The Sylow Theorems
    29.2 Applications of the Sylow Theorems
    29.3 The Fundamental Theorem for Finite Abelian Groups

  • 30. Solvable Groups
    30.1 Solvability
    30.2 New Solvable Groups from Old
    Section VI in a Nutshell

  • VII Unique Factorization

  • 31. Quadratic Extensions of the Integers
    31.1 Quadratic Extensions of the Integers
    31.2 Units in Quadratic Extensions
    31.3 Irreducibles in Quadratic Extensions
    31.4 Factorization for Quadratic Extensions

  • 32. Factorization
    32.1 How Might Factorization Fail?
    32.2 PIDs Have Unique Factorization
    32.3 Primes

  • 33. Unique Factorization
    33.1 UFDs .
    33.2 A Comparison between Z and Z[p??5]
    33.3 All PIDs Are UFDs

  • 34. Polynomials with Integer Coecients
    34.1 The Proof That Q[x] Is a UFD
    34.2 Factoring Integers out of Polynomials
    34.3 The Content of a Polynomial
    34.4 Irreducibles in Z[x] Are Prime

  • 35. Euclidean Domains
    35.1 Euclidean Domains
    35.2 The Gaussian Integers
    35.3 Euclidean Domains Are PIDs
    35.4 Some PIDs Are Not Euclidean
    Section VII in a Nutshell

  • VIII Constructibility Problems

  • 36. Constructions with Compass and Straightedge
    36.1 Construction Problems
    36.2 Constructible Lengths and Numbers

  • 37. Constructibility and Quadratic Field Extensions
    37.1 Quadratic Field Extensions
    37.2 Sequences of Quadratic Field Extensions
    37.3 The Rational Plane
    37.4 Planes of Constructible Numbers
    37.5 The Constructible Number Theorem

  • 38. The Impossibility of Certain Constructions
    38.1 Doubling the Cube
    38.2 Trisecting the Angle
    38.3 Squaring the Circle
    Section VIII in a Nutshell

  • IX Vector Spaces and Field Extensions

  • 39. Vector Spaces I
    39.1 Vectors
    39.2 Vector Spaces

  • 40. Vector Spaces II
    40.1 Spanning Sets
    40.2 A Basis for a Vector Space
    40.3 Finding a Basis
    40.4 Dimension of a Vector Space

  • 41. Field Extensions and Kronecker's Theorem
    41.1 Field Extensions
    41.2 Kronecker
    s Theorem
    41.3 The Characteristic of a Field

  • 42. Algebraic Field Extensions
    42.1 The Minimal Polynomial for an Element
    42.2 Simple Extensions
    42.3 Simple Transcendental Extensions
    42.4 Dimension of Algebraic Simple Extensions

  • 43. Finite Extensions and Constructibility Revisited
    43.1 Finite Extensions
    43.2 Constructibility Problems
    Section IX in a Nutshell

  • X Galois Theory

  • 44. The Splitting Field
    44.1 The Splitting Field
    44.2 Fields with Characteristic Zero

  • 45. Finite Fields
    45.1 Existence and Uniqueness
    45.2 Examples

  • 46 Galois Groups
    46.1 The Galois Group
    46.2 Galois Groups of Splitting Fields

  • 47. The Fundamental Theorem of Galois Theory
    47.1 Subgroups and Sub elds
    47.2 Symmetric Polynomials
    47.3 The Fixed Field and Normal Extensions
    47.4 The Fundamental Theorem
    47.5 Examples

  • 48. Solving Polynomials by Radicals
    48.1 Field Extensions by Radicals
    48.2 Re ning the Root Tower
    48.3 Solvable Galois Groups
    Section X in a Nutshell

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