Math 533: Abstract Algebra I

Professor: Jonah Blasiak

Fall 2016

Mondays and Wednesdays 12:30-1:50pm, One Drexel Plaza 007
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: Korman 275, Tuesday 2:30-4pm, Thursday 1-2:30pm.
Problem Session: Korman Lounge, Tuesday 2-4pm.
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

    Syllabus

    Week 1: Sep 19 (Mon), Sep 21 (Wed)
    Definitions and examples of rings and ideals, isomorphism theorems
    Read 7.1--7.4 in Dummit and Foote
    Problem Set 1, Due Wednesday, Sep 28.

    Week 2: Sep 26 (Mon), Sep 28 (Wed)
    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 Wednesday, Oct 5.

    Week 3: Oct 03, Oct 05
    Module isomorphism theorems, products, coproducts, bases, rank
    Read 10.2--10.3
    Problem Set 3, Due Wednesday, Oct 12.

    Week 4: Oct 12
    Chinese Remainder Theorem, principal ideal domains
    Read 7.6, skim 8.1--8.3
    Problem Set 4, Due Wednesday, Oct 19.

    Week 5: Oct 17, Oct 19
    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--12.3
    Problem Set 5, Due Wednesday, Oct 26.

    Week 6: Oct 24, Oct 26
    Rational canonical form, Jordan canonical form, Representation Theory
    Read 12.1--12.2, 18.1
    Problem Set 6, Due Wednesday, Nov 2.

    Week 7: Oct 31, Nov 02
    Representation Theory, Character Theory
    Read 18.1, 18.3
    Problem Set 7, Due Wednesday, Nov 9.

    Week 8: Nov 07, Nov 09
    Character Theory
    Read 18.3, 19.1
    Problem Set 8, Due Wednesday, Nov 16.

    Week 9: Nov 14, Nov 16
    Artin-Wedderburn Theorem, Determinants
    Read 18.2, 11.4
    Problem Set 9, Due Wednesday, Nov 30.

    Week 10: Nov 21
    Tensor Products
    Read 10.4

    Week 11: Nov 28, Nov 30
    Tensor products, tensor algebra, exterior algebra
    Read 10.4, 11.5

    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.