Math 533: Abstract Algebra I

Professor: Jonah Blasiak

Winter 2018

Tuesdays and Thursdays 3:30-4:50pm, One Drexel Plaza 137

Course Description: Introduction to rings and modules, structure theorem for modules over a principal ideal domain, representation theory, Artin-Wedderburn theorem, tensor products, symmetric and exterior algebras, Schur functions, Schur-Weyl duality.
Prerequisites: at least one semester of abstract algebra.
Level: graduate.
Office Hours: Academic Building 322, Wednesday 1:30pm-3pm.
MRC Hours: Wednesday 3pm-5pm, Thursday 6pm-7pm.
Problem Session: Academic Building 3rd Floor Lounge, Wednesday 2pm-5pm.
Required text: Abstract Algebra, 3rd Edition, David S. Dummit and Richard M. Foote.
Other references:
Homework Policy: You may consult each other and the textbooks above. List all people and sources who aided you and whom you aided, and write up the solutions independently, in your own language. Many exercises will be taken from Dummit and Foote. Solutions to these are available online. Please do not consult these until after you have turned in your homework.
Grading policy:
  • 100% homework
  • Important University Policies:

    Academic Dishonesty

    Disability Resources

    Course Drop Policy

    Code of Conduct


    Week 1: Jan 09 (Tue), Jan 11 (Thu)
    Definitions and examples of rings and ideals, isomorphism theorems
    Read 7.1--7.3 in Dummit and Foote
    Problem Set 1, Due Thursday, Jan 18.

    Week 2: Jan 16 (Tue), Jan 18 (Thu)
    Prime ideals and maximal ideals, definitions and examples of modules
    Read 7.4, 10.1--10.2 in Dummit and Foote
    Problem Set 2, Due Thursday, Jan 25.

    Week 3: Jan 23, Jan 25
    Module isomorphism theorems, products, coproducts, bases, rank
    Read 10.2--10.3
    Problem Set 3, Due Thursday, February 1.
    Problem 9 from Problem Set 2 was moved to Problem 5 on this set.

    Week 4: Jan 30, Feb 01
    Chinese Remainder Theorem, principal ideal domains
    Read 7.6, skim 8.1--8.3
    Problem Set 4, Due Thursday, February 8.

    Week 5: Feb 06, Feb 08
    Structure theorem for modules over a principal ideal domain
    The Wikipedia page on Smith normal form is close to the proof given in class.
    Read 12.1--12.2
    Problem Set 5, Due Thursday, February 15.

    Week 6: Feb 13, Feb 15
    Rational canonical form, Jordan canonical form
    Read 12.2--12.3
    Problem Set 6, Due Thursday, February 22.

    Week 7: Feb 20, Feb 22
    Representation Theory, Character Theory
    Read 18.1, 18.3
    Problem Set 7, Due Thursday, March 1.

    Week 8: Feb 27, Mar 01
    Character Theory
    Read 18.3, 19.1
    Problem Set 8, Due Thursday, March 8.

    Week 9: Mar 06, Mar 08
    Character Theory Read 18.3, 19.1
    Problem Set 9, Due Thursday, March 15.

    Week 10: Mar 13, Mar 15
    Artin-Wedderburn Theorem, Tensor Products
    Read 18.2, 10.4

    We will cover some subset of the following topics, to be refined as the course progresses:

    Definitions and examples of rings and ideals, isomorphism theorems
    Definitions and examples of modules, isomorphism theorems
    Polynomial rings
    Graded rings and modules
    Jordan-Hölder Theorem
    Products, coproducts, universal properties
    Chinese Remainder Theorem
    Structure theorem for modules over a principal ideal domain
    Artin-Wedderburn Theorem
    Introduction to Representation Theory: irreducible representations, Maschke's Theorem, Schur's Lemma, character table, orthogonality relations
    Tensor products, symmetric and exterior algebra, determinants
    Representations of the symmetric group
    Outcomes: The students should have an understanding of the basic objects of abstract algebra including groups, rings, and modules. They should also gain an understanding of homomorphisms, direct sums and products, and tensor products. They should have a knowledge of the basic theorems in this area including isomorphism theorems and universal properties. They should be familiar with representation theory and the structure theorem for modules over a principal ideal domain.