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December 27, 2022


Unlimited Duration


This course includes:

Unlimited Duration

Badge on Completion

Certificate of completion

Unlimited Duration


Algebra II is the second semester of a year-long introduction to modern algebra.

The course focuses on group representations, rings, ideals, fields, polynomial rings, modules, factorization, integers in quadratic number fields, field extensions, and Galois theory.

These notes, which were created by students in a recent on-campus 18.702 Algebra II class, are offered here to supplement the materials included in OCW’s version of 18.702. They have not been checked for accuracy by the instructors of that class or by other MIT faculty members.

Course Curriculum

  • Representations Unlimited
  • Characters and the Direct Sum Unlimited
  • Irreducible Representations Unlimited
  • The Main Theorem Unlimited
  • Characters and Schur’s Lemma Unlimited
  • Orthonormality of Characters Unlimited
  • Orthonormality of Characters Unlimited
  • Rings Unlimited
  • Building New Rings Unlimited
  • Ideals in Polynomial Rings Unlimited
  • More About Rings Unlimited
  • Factorization in Rings Unlimited
  • More Factorization Unlimited
  • Number Fields Unlimited
  • Ideal Factorization Unlimited
  • Uniqueness of Ideal Factorization Unlimited
  • Ideals in Quadratic Fields Unlimited
  • The Ideal Class Group Unlimited
  • Modules over a Ring Unlimited
  • Modules and Presentation Matrices Unlimited
  • Smith Normal Form Unlimited
  • Decomposition of Modules Unlimited
  • Noetherian Rings Unlimited
  • Fields Unlimited
  • Field Extensions Unlimited
  • Finite Fields Unlimited
  • Finite Fields (continued) Unlimited
  • Geometry of Function Fields Unlimited
  • Galois Theory Unlimited
  • Main Theorem of Galois Theory Unlimited
  • Applications of the Galois Correspondence Unlimited
  • Solving Polynomial Equations Unlimited
  • Symmetric Polynomials and the Discriminant Unlimited
  • Solving Polynomial Equations (continued) Unlimited
  • Final Remarks Unlimited
  • Dimensions of Irreducible Characters Unlimited

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Massachusetts Institute of Technology