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Basic Abstract Algebra

Book Description

This book represents a complete course in abstract algebra, providing instructors with flexibility in the selection of topics to be taught in individual classes. All the topics presented are discussed in a direct and detailed manner. Throughout the text, complete proofs have been given for all theorems without glossing over significant details or leaving important theorems as exercises. The book contains many examples fully worked out and a variety of problems for practice and challenge. Solutions to the odd-numbered problems are provided at the end of the book. This new edition contains an introduction to lattices, a new chapter on tensor products and a discussion of the new (1993) approach to the celebrated Lasker-Noether theorem. In addition, there are over 100 new problems and examples, particularly aimed at relating abstract concepts to concrete situations.

Table of Contents

  1. Cover
  2. Title
  3. Copyright
  4. Contents
  5. Preface to the second edition
  6. Preface to the first edition
  7. Glossary of symbols
  8. Part I: Preliminaries
    1. Chapter 1: Sets and mappings
      1. 1 Sets
      2. 2 Relations
      3. 3 Mappings
      4. 4 Binary operations
      5. 5 Cardinality of a set
    2. Chapter 2: Integers, real numbers, and complex numbers
      1. 1 Integers
      2. 2 Rational, real, and complex numbers
      3. 3 Fields
    3. Chapter 3: Matrices and determinants
      1. 1 Matrices
      2. 2 Operations on matrices
      3. 3 Partitions of a matrix
      4. 4 The determinant function
      5. 5 Properties of the determinant function*
      6. 6 Expansion of det A
  9. Part II: Groups
    1. Chapter 4: Groups
      1. 1 Semigroups and groups
      2. 2 Homomorphisms
      3. 3 Subgroups and cosets
      4. 4 Cyclic groups
      5. 5 Permutation groups
      6. 6 Generators and relations
    2. Chapter 5: Normal subgroups
      1. 1 Normal subgroups and quotient groups
      2. 2 Isomorphism theorems
      3. 3 Automorphisms
      4. 4 Conjugacy and G-sets
    3. Chapter 6: Normal series
      1. 1 Normal series
      2. 2 Solvable groups
      3. 3 Nilpotent groups
    4. Chapter 7: Permutation groups
      1. 1 Cyclic decomposition
      2. 2 Alternating group An
      3. 3 Simplicity of An
    5. Chapter 8: Structure theorems of groups
      1. 1 Direct products
      2. 2 Finitely generated abelian groups
      3. 3 Invariants of a finite abelian group
      4. 4 Sylow theorems
      5. 5 Groups of orders p2, pq
  10. Part III: Rings and modules
    1. Chapter 9: Rings
      1. 1 Definition and examples
      2. 2 Elementary properties of rings
      3. 3 Types of rings
      4. 4 Subrings and characteristic of a ring
      5. 5 Additional examples of rings
    2. Chapter 10: Ideals and homomorphisms
      1. 1 Ideals
      2. 2 Homomorphisms
      3. 3 Sum and direct sum of ideals
      4. 4 Maximal and prime ideals
      5. 5 Nilpotent and nil ideals
      6. 6 Zorn’s lemma
    3. Chapter 11: Unique factorization domains and euclidean domains
      1. 1 Unique factorization domains
      2. 2 Principal ideal domains
      3. 3 Euclidean domains
      4. 4 Polynomial rings over UFD
    4. Chapter 12: Rings of fractions
      1. 1 Rings of fractions
      2. 2 Rings with Ore condition
    5. Chapter 13: Integers
      1. 1 Peano’s axioms
      2. 2 Integers
    6. Chapter 14: Modules and vector spaces
      1. 1 Definition and examples
      2. 2 Submodules and direct sums
      3. 3 R-homomorphisms and quotient modules
      4. 4 Completely reducible modules
      5. 5 Free modules
      6. 6 Representation of linear mappings
      7. 7 Rank of a linear mapping
  11. Part IV: Field theory
    1. Chapter 15: Algebraic extensions of fields
      1. 1 Irreducible polynomials and Eisenstein criterion
      2. 2 Adjunction of roots
      3. 3 Algebraic extensions
      4. 4 Algebraically closed fields
    2. Chapter 16: Normal and separable extensions
      1. 1 Splitting fields
      2. 2 Normal extensions
      3. 3 Multiple roots
      4. 4 Finite fields
      5. 5 Separable extensions
    3. Chapter 17: Galois theory
      1. 1 Automorphism groups and fixed fields
      2. 2 Fundamental theorem of Galois theory
      3. 3 Fundamental theorem of algebra
    4. Chapter 18: Applications of Galois theory to classical problems
      1. 1 Roots of unity and cyclotomic polynomials
      2. 2 Cyclic extensions
      3. 3 Polynomials solvable by radicals
      4. 4 Symmetric functions
      5. 5 Ruler and compass constructions
  12. Part V: Additional topics
    1. Chapter 19: Noetherian and artinian modules and rings
      1. 1 HOM R
      2. 2 Noetherian and artinian modules
      3. 3 Wedderburn - Artin theorem
      4. 4 Uniform modules, primary modules, and Noether-Lasker theorem
    2. Chapter 20: Smith normal form over a PID and rank
      1. 1 Preliminaries
      2. 2 Row module, column module, and rank
      3. 3 Smith normal form
    3. Chapter 21: Finitely generated modules over a PID
      1. 1 Decomposition theorem
      2. 2 Uniqueness of the decomposition
      3. 3 Application to finitely generated abelian groups
      4. 4 Rational canonical form
      5. 5 Generalized Jordan form over any field
    4. Chapter 22: Tensor products
      1. 1 Categories and functors
      2. 2 Tensor products
      3. 3 Module structure of tensor product
      4. 4 Tensor product of homomorphisms
      5. 5 Tensor product of algebras
  13. Solutions to odd-numbered problems
  14. Selected bibliography
  15. Index