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Algebra: Abstract and Modern

Book Description

Algebra: Abstract and Modern, spread across 16 chapters, introduces the reader to the preliminaries of algebra and then explains topics like group theory and field theory in depth. It also features a blend of numerous challenging exercises and examples that further enhance each chapter. Covering all the necessary topics on the subject, this text is an ideal text book for an undergraduate course on mathematics.

Table of Contents

  1. Cover
  2. Title Page
  3. Contents
  4. Dedication
  5. Preface
  6. Part I: Preliminaries
    1. 1. Sets and Relations
      1. 1.1 Sets and subsets
      2. 1.2 Relations and functions
      3. 1.3 Equivalence relations and partitions
      4. 1.4 The cardinality of a set
    2. 2. Number Systems
      1. 2.1 Integers
      2. 2.2 Congruence modulo n
      3. 2.3 Rational, real and complex numbers
      4. 2.4 Ordering
      5. 2.5 Matrices
      6. 2.6 Determinants
  7. Part II: Group Theory
    1. 3. Groups
      1. 3.1 Binary systems
      2. 3.2 Groups
      3. 3.3 Elementary properties of groups
      4. 3.4 Finite groups and group tables
    2. 4. Subgroups and Quotient Groups
      1. 4.1 Subgroups
      2. 4.2 Cyclic groups
      3. 4.3 Cosets of a subgroup
      4. 4.4 Lagrange’s theorem
      5. 4.5 Normal subgroups
      6. 4.6 Quotient groups
    3. 5. Homomorphisms of Groups
      1. 5.1 Definition and examples
      2. 5.2 Fundamental theorem of homomorphisms
      3. 5.3 Isomorphism theorems
      4. 5.4 Automorphisms
    4. 6. Permutation Groups
      1. 6.1 Cayley’s theorem
      2. 6.2 The symmetric group Sn
      3. 6.3 Cycles
      4. 6.4 Alternating group An and dihedral group Dn
    5. 7. Group Actions on Sets
      1. 7.1 Action of a group on a set
      2. 7.2 Orbits and stabilizers
      3. 7.3 Certain counting techniques
      4. 7.4 Cauchy and Sylow theorems
    6. 8. Structure Theory of Groups
      1. 8.1 Direct products
      2. 8.2 Finitely generated abelian groups
      3. 8.3 Invariants of finite abelian groups
      4. 8.4 Groups of small order
  8. Part III: Ring Theory
    1. 9. Rings
      1. 9.1 Examples and elementary properties
      2. 9.2 Certain special elements in rings
      3. 9.3 The characteristic of a ring
      4. 9.4 Subrings
      5. 9.5 Homomorphisms of rings
      6. 9.6 Certain special types of rings
      7. 9.7 Integral domains and fields
    2. 10. Ideals and Quotient Rings
      1. 10.1 Ideals
      2. 10.2 Quotient rings
      3. 10.3 Chinese remainder theorem
      4. 10.4 Prime ideals
      5. 10.5 Maximal ideals
      6. 10.6 Embeddings of rings
    3. 11. Polynomial Rings
      1. 11.1 Rings of polynomials
      2. 11.2 The division algorithm
      3. 11.3 Polynomials over a field
      4. 11.4 Irreducible polynomials
    4. 12. Factorization in Integral Domains
      1. 12.1 Divisibility in integral domains
      2. 12.2 Principal ideal domains
      3. 12.3 Unique factorization domains
      4. 12.4 Polynomials over UFDs
      5. 12.5 Euclidean domains
      6. 12.6 Some applications to number theory
    5. 13. Modules and Vector Spaces
      1. 13.1 Modules and submodules
      2. 13.2 Homomorphisms and quotients of modules
      3. 13.3 Direct products and sums
      4. 13.4 Simple and completely reducible modules
      5. 13.5 Free modules
      6. 13.6 Vector spaces
  9. Part IV: Field Theory
    1. 14. Extension Fields
      1. 14.1 Extensions of a field
      2. 14.2 Algebraic extensions
      3. 14.3 Algebraically closed fields
      4. 14.4 Derivatives and multiple roots
      5. 14.5 Finite fields
    2. 15. Galois Theory
      1. 15.1 Separable and normal extensions
      2. 15.2 Automorphism groups and fixed fields
      3. 15.3 Fundamental theorem of Galois theory
    3. 16. Selected Applications of Galois Theory
      1. 16.1 Fundamental theorem of algebra
      2. 16.2 Cyclic extensions
      3. 16.3 Solvable groups
      4. 16.4 Polynomials solvable by radicals
      5. 16.5 Constructions by ruler and compass
  10. Answers/Hints to Selected Even-Numbered Exercises
  11. Copyright