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Abstract Algebra by Irena Swanson

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Abstract Algebra written by Irena Swanson . This is an other great mathematics book cover the following topics of Problem Solving.

  • Definition and examples
  • What follows immediately from the definition
  • Bijections
  • Commutativity
  • Frequent groups and groups with names
  • Group generators
  • Subgroups
  • Plane groups
  • Orders of groups and elements
  • One-generated subgroups
  • The Euler φ function – an aside
  • Permutation groups
  • Group homomorphisms
  • Group isomorphisms
  • Group actions
  • Cosets and Lagrange’s Theorem
  • RSA public key encryption scheme
  • Stabilizers, orbits
  • Centralizer and the class equation
  • External direct products/sums
  • Normal subgroups
  • Factor (or quotient) groups
  • The internal direct product
  • The isomorphism theorems
  • Fundamental Theorem of Finite Abelian Groups
  • Sandpile groups
  • Rings
  • Some unsurprising definitions
  • Something new: ideals
  • More that’s new: characteristic of a ring
  • Quotient (or factor) rings
  • (Integral) Domains
  • Prime ideals and maximal ideals
  • Division algorithm in polynomial rings
  • Irreducibility
  • Unique factorization domains
  • Monomial orderings (all in exercises)
  • Modules
  • Finitely generated modules over principal ideal domains
  • The Chinese Remainder Theorem
  • Fields
  • Splitting fields
  • Derivatives in algebra (optional)
  • Finite fields
  • Appendix: Euclidean algorithm for integers
  • Work out the RSA algorithm another time
  • Appendix: Going overboard with factoring of X3
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