MAT5173

Introduction to groups rings and fields.

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