# Math 533: Abstract Algebra I

### Winter 2015

Tuesdays and Thursdays 9:30-10:50am, Korman 245

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 2:30-3:30pm, Wednesday 1-3pm.
Problem Session: Korman 207A, Monday 3-5pm.
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
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## Syllabus

Week 1: Jan 06 (Tue), Jan 08 (Thu)
Definitions and examples of rings and ideals, isomorphism theorems
Read 7.1--7.4 in Dummit and Foote
Problem Set 1, Due Tuesday, Jan 13. Problem 8 will be moved to the next problem set since we have not yet discussed prime ideals. Definition needed for Problem 2 (d) (see 10.1): Let F be a field. An F-algebra is a ring A together with a (unital) ring homomorphism f: F --> A such that the image f(F) is contained in the center of A. The center of a ring A is the set {a in A : ab = ba for every b in A}. It follows from this definition that an F-algebra is also an F-vector space. So a finite-dimensional F-algebra is an F-algebra that is also finite-dimensional as an F-vector space.

Week 2: Jan 13 (Tue), Jan 15 (Thu)
Prime ideals and maximal ideals, definitions and examples of modules, isomorphism theorems
Read 7.4, 10.1--10.2 in Dummit and Foote
Problem Set 2, Due Tuesday, Jan 20.

Week 3: Jan 20, Jan 22
Products, coproducts, Chinese Remainder Theorem, principal ideal domains
Read 10.3, 7.6, 12.1, skim 8.1--8.3
Problem Set 3, Due Tuesday, Jan 27.

Week 4: Jan 27, Jan 29
Structure theorem for modules over a principal ideal domain, determinants
The Wikipedia page on Smith normal form is close to the proof given in class.
Problem Set 4, Due Tuesday, Feb 3.
Correction: Problem 8 is false! So, instead of proving that x_1,...,x_n form a basis, find an example of a commutative ring R and linearly independent elements x_1,...,x_n of R^n such that these elements do not form a basis of R^n.
Problem 9 is correct.

Week 5: Feb 03, Feb 05
Rational canonical form, Jordan canonical form
Problem Set 5, Due Tuesday, Feb 10.

Week 6: Feb 10, Feb 12
Representation Theory
Problem Set 6, Due Tuesday, Feb 17.
Note: For Problem 9, work over the complex numbers, i.e., determine the representations D_4n -> GL_1(C).

Week 7: Feb 17, Feb 19
Representation Theory, Character Theory
Problem Set 7, Due Tuesday, Feb 24.

Week 8: Feb 24, Feb 26
Character Theory
Problem Set 8, Due Tuesday, Mar 03.

Week 9: Mar 03, Mar 05
Artin-Wedderburn Theorem, Tensor Products
Problem Set 9, Due Tuesday, Mar 10.

Week 10: Mar 10, Mar 12
Tensor algebra, exterior algebra
Problem Set 10, do not turn in.

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
R-algebras
Graded rings and modules
Jordan-Hölder 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
Localization
Direct product, direct sum, universal properties
Representations of the symmetric group
Representations of GL_n, schur functions, Schur-Weyl duality
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.