# Math 533: Abstract Algebra I

### Winter 2019

Tuesdays and Thursdays 3:30-4:50pm, One Drexel Plaza GL41

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.
Office Hours: Korman 275, Tuesday 2pm-3pm, Wednesday 2:30pm-4pm.
Problem Session: Korman 245, Wednesday 2pm-5pm.
Required text: Abstract Algebra, 3rd Edition, David S. Dummit and Richard M. Foote.
Other references:
• The Symmetric Group: Representations, Combinatorial Algorithms, and Symmetric Functions, 2nd Edition, Bruce E. Sagan.
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.
• 100% homework

• ## Syllabus

Week 1: Jan 08 (Tue), Jan 10 (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 17.

Week 2: Jan 15 (Tue), Jan 17 (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 24.

Week 3: Jan 22, Jan 24
Module isomorphism theorems, products, coproducts, bases, rank
Problem Set 3, Due Thursday, Jan 31.
Problem 9 from Problem Set 2 was moved to Problem 5 on this set.

Week 4: Jan 29, Jan 31
Chinese Remainder Theorem, principal ideal domains
Problem Set 4, Due Thursday, February 7.

Week 5: Feb 05, Feb 07
Structure theorem for modules over a principal ideal domain
The Wikipedia page on Smith normal form is close to the proof given in class.
Problem Set 5, Due Thursday, February 14.

Week 6: Feb 12, Feb 14
Rational canonical form, Jordan canonical form
Problem Set 6, Due Tuesday, February 26.
There is an extension for Problem set 6. It is now due February 26. Problem set 7 will still be due February 28.

Week 7: Feb 19, Feb 21
Representation Theory
Problem Set 7, Due Thursday, February 28.

Week 8: Feb 26, Feb 28
Character Theory
Problem Set 8, Due Thursday, March 7.

Week 9: Mar 05, Mar 07
Character Theory
Problem Set 9, Due Thursday, March 14.

Week 10: Mar 12, Mar 14
Artin-Wedderburn Theorem

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
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
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