Algebra I

Department: 
MATH
Course Number: 
6121
Hours - Lecture: 
3
Hours - Lab: 
0
Hours - Recitation: 
0
Hours - Total Credit: 
3
Typical Scheduling: 
Every fall semester

Graduate level linear and abstract algebra including groups, rings, modules, and fields. (1st of two courses)

Prerequisites: 

MATH 4107 and one of MATH 2406, MATH 4305, or permission of instructor

Course Text: 

Text at the level of Abstract Algebra by Dummit and Foote.

Topic Outline: 
  • Intensive review of elementary group theory: groups, subgroups, homomorphisms, quotient groups, Lagrange's theorem, permutation groups
  • Group actions, Burnside's Lemma
  • The Class Equation, the Sylow theorems
  • Simple groups and composition series
  • Free groups, generators and relations
  • Direct and semidirect products
  • Structure theorem for finitely generated abelian groups
  • Rings, ideals, quotient rings
  • The Chinese Remainder Theorem
  • Euclidean domains, Principal Ideal Domains, Unique Factorization Domains
  • Polynomial rings
  • Modules, submodules, quotient modules, free modules
  • Finitely generated modules over a Principal Ideal Domain
  • Rational and Jordan Canonical Forms
  • Fields, algebraic and transcendental extensions
  • Splitting fields, algebraic closure
  • Finite fields
  • Separable and inseparable extensions
  • Classical straightedge and compass constructions