This book is on abstract algebra (abstract algebraic systems), an advanced set of topics related to algebra, including groups, rings, ideals, fields, and more. Readers of this book are expected to have read and understood the information presented in the Linear Algebra book, or an equivalent alternative.

Table of Contents

  1. Introduction
    1. Sets
    2. Equivalence relations and congruence classes
    3. Functions
    4. Binary Operations
    5. Linear Algebra
    6. Number Theory
  2. Group Theory
    1. Groups
    2. Subgroups
    3. Cyclic groups
    4. Permutation groups
    5. Homomorphism
    6. Normal subgroups and Quotient groups
    7. Products and Free groups
    8. Group actions on sets
    9. Composition series
    10. The Sylow Theorems
  3. Rings
    1. Rings
    2. Ring Homomorphisms
    3. Ideals
    4. Integral domains
    5. Fraction Fields
    6. Polynomial Rings
  4. Fields
    1. Fields
    2. Factorization
    3. Splitting Fields and Algebraic Closures
    4. Separability, Normal Extensions
  5. Vector Spaces
    1. Vector Spaces
  6. Modules
    1. Modules
    2. Hypercomplex numbers
  7. Algebras
    1. Algebras
    2. Boolean algebra
    3. Clifford Algebras
    4. Quaternions
    5. 2x2 real matrices
    6. Galois Theory
  8. Further abstract algebra
    1. Category theory
    2. Lattice theory
    3. Matroids
  9. Authors

Information for contributors

This wikibook shall give an introduction to the fundamental concepts of abstract algebra, such as groups, rings and ideals, and fields and Galois theory.

Contents

  1. Definition of groups, very basic properties

Groups

  1. Groups and subgroups
  2. Abelian groups
  3. Representations
  4. p-groups

Rings

  1. Rings, ideals, ring homomorphisms
  2. The hierarchy of rings
  3. Polynomial rings, irreducibility
  4. Fields of fractions

Fields

  1. Fields and prime fields
  2. Algebraic field extensions
  3. Elementary Galois theory
  4. Transcendental extensions

Sources
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