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Introduction to groups and rings.

Sample textbook

[1] Thomas W. Judson and Robert A. Beezer, Abstract Algebra: Theory and Applications, 2008. Freely available online.

Catalog entry

Prerequisite: Algebra and Number Systems (MAT 1313), or Discrete Mathematical Structures (CS 2233/2231), or instructor consent.

Contents (1) Groups: Cyclic groups, permutation groups and Cayley's theorem, group homomorphisms, normal subroups, quotient groups and Lagrange's theore, the theorems of Euler and Fermat. (2) Rings: Ring homomorphisms, integral domains and fields, maximal and prime ideals. (3) Rings of polynomials: The Division Algorithm and irreducible polynomials.

Topics List

Week Topic Sections from the Judson-Beezer book Subtopics Prerequisite
1-2 Groups 3
  • Definitions and classical examples
  • Subgroups
  • Isomorphisms
MAT1313, CS2233/2231, or instructor consent.
4-5 Cyclic groups 4
  • Classification of cyclic groups.
5-6 Permutation groups 5
  • Permutations
  • Cayley's Theorem
7-8 Cosets and Lagrange's Theorem 10
  • Normal subgroups
  • Factor Groups
  • The theorems of Euler and Fermat
9 Homomorphisms 11 The Isomorphism Theorem
10-11 Rings 16
  • Ring homomorphisms
  • Integral domains and fields
  • Maximal and Prime Ideals
12-end Rings of Polynomials 17
  • The Division Algorithm
  • Irreducible Polynomials
  • Solving cubic and quartic equations