Abstract Algebra

Abstract algebra (also known as modern algebra) is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras.

In an introductory course on abstract algebra and its application, one may teach the following topics:

  • Definition of binary operations

  • Definition of semigroups, monoids, and groups

  • Subgroups

  • Normal subgroups and quotient groups

  • Group homomorphisms

  • Automorphisms

  • Cayley's theorem

  • Sylow's theorems

  • Definition of rings

  • Ring homomorphisms

  • Ideals and quotient rings

  • Prime and maximal ideals of rings

  • The field of quotients of an integral domain

  • Euclidean rings

  • Polynomial and formal power series rings

For more details, refer to the following book:

Herstein, I. N. (2006). Topics in algebra. John Wiley & Sons.

For applications of abstract algebra, one may choose some topics from the following general themes:

  • Coding Theory

  • Cryptography

For more details, refer to the following book:

Lidl, R., Pilz, G. (2012). Applied Abstract Algebra. Springer Science & Business Media.

See also the following pages: