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Abstract Algebra Theory and Applications by Thomas W. Judson

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Abstract Algebra Theory and Applications written by Thomas W. Judson . This is an other great mathematics book cover the following topics of Problem Solving.

  • Preliminaries
    A Short Note on Proofs, Sets and Equivalence Relations

  • The Integers
    Mathematical Induction, The Division Algorithm

  • Groups
    Integer Equivalence Classes and Symmetries, Definitions and Examples, Subgroups

  • Cyclic Groups
    Cyclic Subgroups, Multiplicative Group of Complex Numbers, The Method of Repeated Squares

  • Permutation Groups
    Definitions and Notation, Dihedral Groups

  • Cosets and Lagrange’s Theorem
    Cosets, Lagrange’s Theorem, Fermat’s and Euler’s Theorems

  • Introduction to Cryptography
    Private Key Cryptography, Public Key Cryptography

  • Algebraic Coding Theory
    Error-Detecting and Correcting Codes, Linear Codes, Parity-Check and Generator Matrices, Efficient Decoding

  • Isomorphisms
    Definition and Examples, Direct Products

  • Normal Subgroups and Factor Groups
    Factor Groups and Normal Subgroups, The Simplicity of the Alternating Group

  • Homomorphisms
    Group Homomorphisms, The Isomorphism Theorems

  • Matrix Groups and Symmetry
    Matrix Groups, Symmetry

  • The Structure of Groups
    Finite Abelian Groups, Solvable Groups

  • Group Actions
    Groups Acting on Sets, The Class Equation, Burnside’s Counting Theorem

  • The Sylow Theorems
    The Sylow Theorems, Examples and Applications

  • Rings
    Rings, Integral Domains and Fields, Ring Homomorphisms and Ideals, Maximal and Prime Ideals, An Application to Software Design

  • Polynomials
    Polynomial Rings, The Division Algorithm, Irreducible Polynomials

  • Integral Domains
    Fields of Fractions, Factorization in Integral Domains

  • Lattices and Boolean Algebras
    Lattices, Boolean Algebras, The Algebra of Electrical Circuits

  • Vector Spaces
    Definitions and Examples, Subspaces, Linear Independence

  • Fields
    Extension Fields, Splitting Fields, Geometric Constructions

  • Finite Fields
    Structure of a Finite Field, Polynomial Codes

  • Galois Theory
    Field Automorphisms, The Fundamental Theorem, Applications

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