COURSE NUMBER AND TITLE: MATH 4212 Modern Abstract Algebra II

CREDIT HOURS: 3

CATALOG DESCRIPTION: A continuation of the study of abstract algebraic structure. Topics include rings, ideals, integral domains, fields, and rings of polynomials.

PREREQUISITE(S): MATH 4211

SUGGESTED TEXT(S):
A First Course in Abstract Algebra by John Fraleigh
Contemporary Abstract Algebra by Joseph Gallian
A Book of Abstract Algebra by Charles Pinter

COURSE OUTLINE:

1. Basic properties of rings
   • The ring axioms and examples of rings
   • Zero-divisors and units of a ring
   • Integral domains and fields
   • The field of quotients of an integral domain
   • Polynomial rings and the evaluation homomorphism
   • The division algorithm and the factor theorem in F[x]
   • Irreducibility and Eisenstein's criterion
   • Unique factorization in F[x]
2. More advanced ring theory
   • Examples of noncommutative rings
   • Ring homomorphisms and analogues of results from group theory
   • Direct products of rings
   • Ideals and quotient rings
   • The First Isomorphism Theorem for rings
   • Prime and maximal ideals
   • UFD's, PID's, and Euclidean domains
3. Fields
   • Construction of finite fields
   • Extension fields
   • Algebraic and transcendental field extensions
   • Kronecker's theorem
   • Algebraic closure
   • An introduction to Galois theory
4. Other topics (as time permits)
   • The ring of quaternions
   • Chain conditions (Artinian/Noetherian rings)
   • Classical applications of Galois theory (geometric constructions)