Algebra I

Course Number: 
Hours - Lecture: 
Hours - Lab: 
Hours - Recitation: 
Hours - Total Credit: 
Typical Scheduling: 
Every fall semester

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


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