Abstract algebra bears little resemblance to ordinary algebra, which we all studied in high school. They are as different as night and day. Abstract algebra is actually an advanced topic in mathematics that deals with the following topics:

  • Integers and equivalence relations
  • Groups
  • Finite groups; subgroups
  • Cyclic groups
  • Permutation groups
  • Isomorphisms
  • Cosets and Lagrange's Theorem
  • External direct products
  • Normal subgroups and factor groups
  • Group homomorphisms
  • Fundamental theorem of finite abelian groups
  • Rings
  • Subrings
  • Integral domains
  • Ideals and factor rings
  • Ring homomorphisms
  • Polynomial rings
  • Factorization of polynomials
  • Divisibility in integral domains
  • Fields
  • Vector spaces
  • Extension fields
  • Algebraic extensions
  • Finite fields
  • Geometric constructions
  • Sylow Theorems
  • Conjugacy classes
  • Finite simple groups
  • Generators and relations
  • Symmetry groups
  • Symmetry and counting
  • Cayley Digraphs of Groups
  • Algebraic coding theory
  • Galois theory
  • Cyclotomic extensions

A superb website for content related to abstract algebra is called Mathematics Archives. You should also stay up to date with the latest research by reading the Journal of Algebra and Its Applications.

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