Math 418. Intro to Abstract Algebra II
Text: Nicholson, Introduction to Abstract Algebra, 3rd edition
(a) Ring Theory II
Ring of quotients of a domain. Euclidean domains and principal ideal domains. Unique factorization domains. Prime ideals. Irreducible polynomials. Eisenstein's criterion. Gauss's lemma. Unique factorization in polynomial rings over fields. 
(b) Field Theory
Brief review of vector spaces, basis and dimension. Field extensions. Splitting fields of polynomials. Application to ruler and compass constructions 
(c) Modules over Principal Ideal Domains
Introduction to module theory: submodules, quotient modules, homomorphisms of modules. Primary Decomposition Theorem. Free modules. Structure theorem for principal ideal domains. Application to finitely generated abelian groups: structure of groups given by generators and relations using matrix methods. Application to canonical forms: rational form and Jordan form. 
(d) Introduction to Error Correcting Linear Codes
Examples of codes. Hamming distance. Upper and lower bounds for size of a code. Check matrix and generator matrix. 
Exams and leeway = 4 hours. Total = 43 hours.
Grading: Homework = 25%, exams = 35% , final = 40% .
Prerequisite: Math 417.