# Syllabus Math 418

## Math 418. Intro to Abstract Algebra II Instructor Syllabus

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. [12]

(b) Field Theory
Brief review of vector spaces, basis and dimension. Field extensions. Splitting fields of polynomials. Application to ruler and compass constructions [12]

(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. [10]

(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. [5]

Exams and leeway = 4 hours. Total = 43 hours.
Grading: Homework = 25%, exams = 35% , final = 40% .

Prerequisite: Math 417.