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Abstract Algebra (Third Edition) by David S. Dummit and Richard M. Foote



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Abstract Algebra (Third Edition) written by David S. Dummit and Richard M. Foote , University of Vermont.

Abstract Algebra (Third Edition) written by David S. Dummit and Richard M. Foote cover the following topics.

  • 0. Preliminaries

  • 0.1 Basics
    0.2 Properties of the Integers
    0.3 Z In Z : The Integers Modulon
  • Part 1 - GROUP THEORY

  • 1. Introduction to Groups

    1.1 Basic Axioms and Examples
    1.2 Dihedral Groups
    1.3 Symmetric Groups
    1.4 Matrix Groups
    1.5 The Quaternion Group
    1.6 Homomorphisms and Isomorphisms
    1.7 Group Actions


  • 2. Subgroups

    2.1 Definition and Examples
    2.2 Centralizers and Normalizers, Stabilizers and Kernels
    2.3 Cyclic Groups and Cyclic Subgroups
    2.4 Subgroups Generated by Subsets of a Group
    2.5 The Lattice of Subgroups of a Group


  • 3. Quotient Groups and Homomorphisms

    3.1 Definitions and Examples
    3.2 More on Cosets and Lagrange's Theorem
    3.3 The Isomorphism Theorems
    3.4 Composition Series and the Holder Program
    3.5 Transpositions and the Alternating Group


  • 4. Group Actions

    4.1 Group Actions and Permutation Representations
    4.2 Groups Acting on Themselves by Left Multiplication-cayley's Theorem
    4.3 Groups Acting on Themselves by Conjugation-The Class Equation
    4.4 Automorphisms
    4.5 The Sylow Theorems
    4.6 The Simplicity of An


  • 5. Direct and Semidirect Products and Abelian Groups

    5.1 Direct Products
    5.2 The Fundamental Theorem of Finitely Generated Abelian Groups
    5.3 Table of Groups of Small Order
    5.4 Recognizing Direct Products
    5.5 Semidirect Products


  • 6. Further Topics in Group Theory

    6.1 p-groups, Nilpotent Groups, and Solvable Groups Applications in Groups of Medium Order
    6.2 Application
    6.3 A Word on Free Groups

  • Part II - RING THEORY


  • 7. Introduction to Rings

    7.1 Basic Definitions and Examples
    7.2 Examples: Polynomial Rings, Matrix Rings, and Group Rings
    7.3 Ring Homomorphisms an Quotient Rings
    7.4 Properties of Ideals
    7.5 Rings of Fractions
    7.6 The Chinese Remainder Theorem


  • 8. Euclidean Domains, Principal Ideal Domains and Unique Factorization Domains

    8.1 Euclidean Domains
    8.2 Principal Ideal Domains (P.I.D.s}
    8.3 Unique Factorization Domains (U.F.D.s}


  • 9. Polynomial Rings

    9.1 Definitions and Basic Properties
    9.2 Polynomial Rings over Fields I
    9.3 Polynomial Rings that are Unique Factorization Domains
    9.4 Irreducibility Criteria
    9.5 Polynomial Rings over Fields II
    9.6 Polynomials in Several Variables over a Field and Grobner Bases

  • Part III - MODULES AN D VECTOR SPACES

  • 10. Introduction to Module Theory

    10.1
    10.1 Basic Definitions and Examples
    10.2 Quotient Modules and Module Homomorphisms
    10.3 Generation of Modules, Direct Sums, and Free Modules
    10.4 Tensor Products of Modules
    10.5 Exact Sequences-Projective, Injective, and Flat Modules


  • 11. Vector Spaces

    11.1 Definitions and Basic Theory
    11.2 The Matrix of a Linear Transformation
    11.3 Dual Vector Spaces
    11.4 Determinants
    11.5 Tensor Algebras. Symmetric and Exterior Algebras


  • 12. Modules over Principal Ideal Domains

    12.1 The Basic Theory
    12.2 The Rational Canonical Form
    12.3 The jordan Canonical Form

  • Part IV - FIELD THEORY AND GALOIS THEORY

  • 13. Field Theory

    13.1 Basic Theory of Field Extensions
    13.2 Algebraic Extensions
    13.3 Classical Straightedge and Compass Constructions
    13.4 Splitting Fields and Algebraic Closures
    13.5 Separable and Inseparable Extensions
    13.6 Cyclotomic Polynomials and Extensions


  • 14. Galois Theory

    14.1 Basic Definitions
    14.2 The Fundamental Theorem of Galois Theory
    14.3 Finite Fields
    14.4 Composite Extensions and Simple Extensions
    14.5 Cyclotomic Extensions and Abelian Extensions over Q
    14.6 Galois Groups of Polynomials
    14.7 Solvable and Radical Extensions: lnsolvability ofthe Quintic
    14.8 Computation of Galois Groups over Q
    14.9 Transcendental Extensions, Inseparable Extensions, Infinite Galois Groups

  • Part V - AN INTRODUCTION TO COMMUTATIVE RINGS, ALGEBRAIC GEOM ETRY, AND HOMOLOGICAL ALGEBRA

  • 15. Commutative Rings and Algebraic Geometry

    15.1 Noetherian Rings and Affine Algebraic Sets
    15.2 Radicals and Affine Varieties
    15.3 Integral Extensions and Hilbert's Nullstellensatz
    15.4 Localization
    15.5 The Prime Spectrum of a Ring


  • 16. Artinian Rings, Discrete Valuation Rings, and Dedekind Domains

    16.1 Artinian Rings
    16.2 Discrete Valuation Rings
    16.3 Dedekind Domains


  • 17. Introduction to Homological Algebra and Group Cohomology

    17.1 Introduction to Homological Algebra-Ext and Tor
    17.2 The Cohomology of Groups
    17.3 Crossed Homomorphisms and H1(G, A)
    17.4 Group Extensions, Factor Sets and H2(G, A)

  • Part VI - INTRODUCTION TO THE REPRESENTATION THEORY OF FINITE GROUPS

  • 18. Representation Theory and Character Theory

    18.1 Linear Actions and Modules over Group Rings
    18.2 Wedderburn's Theorem and Some Consequences
    18.3 Character Theory and the Orthogonality Relations


  • 19. Examples and Applications of Character Theory

    19.1 Characters of Groups of Small Order
    19.2 Theorems of Burnside and Hall
    19.3 Introduction to the Theory of Induced Characters

  • Appendix 1: Cartesian Products and Zorn's Lemma

  • Appendix II: Category Theory

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