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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 cover the following topics.

  • 1. Preliminaries (A Short Note on Proofs, Sets and Equivalence Relations)

  • 2. The Integers
    Mathematical Induction, The Division Algorithm

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

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

  • 5. Permutation Groups
    Definitions and Notation, Dihedral Groups

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

  • 7. Introduction to Cryptography
    Private Key Cryptography, Public Key Cryptography

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

  • 9. Isomorphisms
    Definition and Examples, Direct Products

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

  • 11. Homomorphisms
    Group Homomorphisms, The Isomorphism Theorems

  • 12. Matrix Groups and Symmetry
    Matrix Groups, Symmetry

  • 13. The Structure of Groups
    Finite Abelian Groups, Solvable Groups

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

  • 15. The Sylow Theorems
    The Sylow Theorems, Examples and Applications

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

  • 17. Polynomials
    Polynomial Rings, The Division Algorithm, Irreducible Polynomials,

  • 18. Integral Domains
    Fields of Fractions, Factorization in Integral Domains

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

  • 20. Vector Spaces
    Definitions and Examples, Subspaces, Linear Independence

  • 21. Fields
    Extension Fields, Splitting Fields, Geometric Constructions

  • 22. Finite Fields
    Structure of a Finite Field, Polynomial Codes

  • 23. Galois Theory
    Field Automorphisms, The Fundamental Theorem, Applications

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