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Abstract Algebra An Introduction with Applications (2nd Edition) by Derek John Scott Robinson



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Abstract Algebra An Introduction with Applications (2nd Edition) written by Derek John Scott Robinson . This is the second edition of the introduction to abstract algebra. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. There is ample material here for a two semester course in abstract algebra. Abstract: This is the second edition of the introduction to abstract algebra. In addition to introducing the main concepts of modern algebra, the book contains numerous applications, which are intended to illustrate the concepts and to convince the reader of the utility and relevance of algebra today. There is sample material here for a two semester course in abstract algebra

Abstract Algebra An Introduction with Applications (2nd Edition) written by Derek John Scott Robinson cover the following topics.

  • 1. Sets, relations and functions
    1.1 Sets and subsets
    1.2 Relations, equivalence relations, partial orders
    1.3 Functions
    1.4 Cardinality

  • 2. The integers
    2.1 Well-ordering and mathematical induction
    2.2 Division in the integers
    2.3 Congruences

  • 3. Introduction to groups
    3.1 Permutations
    3.2 Semigroups, monoids and groups
    3.3 Groups and subgroups

  • 4. Quotient groups and homomorphisms
    4.1 Cosets and Lagrange’s Theorem
    4.2 Normal subgroups and quotient groups
    4.3 Homomorphisms

  • 5. Groups acting on sets
    5.1 Group actions
    5.2 Orbits and stabilizers
    5.3 Applications to the structure of groups
    5.4 Applications to combinatorics

  • 6. Introduction to rings
    6.1 Elementary properties of rings
    6.2 Subrings and ideals
    6.3 Integral domains, division rings and eld
    6.4 Finiteness conditions on ideals

  • 7. Division in commutative rings
    7.1 Euclidean domains
    7.2 Principal ideal domains
    7.3 Unique factorization in integral domains
    7.4 Roots of polynomials and splitting eld

  • 8. Vector spaces
    8.1 Vector spaces and subspaces
    8.2 Linear independence, basis and dimension
    8.3 Linear mappings
    8.4 Eigenvalues and eigenvectors

  • 9. Introduction to modules
    9.1 Elements of module theory
    9.2 Modules over principal ideal domains
    9.3 Applications to linear operators

  • 10. The Structure of groups
    10.1 The Jordan–Hölder Theorem
    10.2 Solvable and nilpotent groups
    10.3 Theorems on nite solvable groupsÊ

  • 11. The Theory of elds
    11.1 Field extensions
    11.2 Constructions with ruler and compass
    11.3 Finite elds
    11.4 Latin squares and Steiner triple systems

  • 12. Galois Theory
    12.1 Normal and separable extensions
    12.2 Automorphisms of eld extensions
    12.3 The Fundamental Theorem of Galois theory
    12.4 Solvability of equations by radicals

  • 13. Tensor products
    13.1 Denition of the tensor product
    13.2 Properties of tensor products.
    13.3 Extending the ring of operators.

  • 14. Further topics
    14.1 Zorn’s Lemma with applications
    14.2 Roots of polynomials and discriminants
    14.3 Presentations of groups
    14.4 Introduction to error correcting codes


  • Bibliography
    List of symbols
    Index

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