MAT 534: Algebra I (Fall 2025)
Announcements
- Office hours changed to Tuesday, 12:30–1:30pm.
- The Midterm will be held in class on Wednesday, October 8. The topics are group theory and category theory.
- Homework 1 is due on Wednesday, September 10 (changed).
About the course
Abstract algebra is the study of algebraic structures such as groups, rings, fields, or vector spaces. In the course, we will study groups, rings and modules, and basic linear algebra. You can find a more detailed list of topics on this page.
Textbook
The textbook for the course is “Abstract Algebra” (3rd edition) by David S. Dummit and Richard M. Foote. Please see the official syllabus for additional information about the course, including university-wide policies.
Grading
The final exam for the course will be on December 17. We are also going to have one midterm (in class) and weekly homework assignments. Your grade will be determined based on the final exam (40%), the midterm (30%), and your homework (30%). The general policy is no make-up exams and no late homework, but there will be an extra homework assignment at the end of the semester to make up for any missed assignments.
Time and location
We meet on Monday and Wednesday, 11:00am–12:20pm, in room Physics P-127 (in the Physics building).
Office hours
My office hours are Tuesday 12:30pm to 1:30pm, Friday 10am to 12pm, and by appointment.
Schedule
Please read the corresponding sections before or after class.
| Week | Chapters | Topics |
| 1 | I.1 and I.2 | Groups, subgroups, examples |
| 2 | I.3 and some of I.4 | Factor groups, isomorphism theorems, group actions |
| 3 | I.4 and some of I.5 | Sylow theorems, applications, direct products, symmetric groups |
| 4 | Some of I.5 | Automorphisms, An is simple, solvable groups |
| 5 | Some of I.6 | Finitely generated abelian groups |
| 6 | Appendix II | Category theory |
| 7 | Some of II.7 | Rings, ideals, examples |
| 8 | II.7 | Integral domains, maximal/prime ideals, fractions fields |
| 9 | II.8 | PIDs, unique factorization, UFDs |
| 10 | Some of II.9 | Gauss lemma, Noetherian rings, Hilbert basis theorem |
| 11 | Some of II.10 | Modules, submodules, examples |
| 12 | Modules over PIDs | |
| 13 | Modules over PIDs, torsion modules, vector spaces | |
| 14 | Modules over PIDs, linear transformations, Jordan canonical form | |
| 15 | Multilinear algebra, determinant, inner products, normal operators |
Homework assignments
There will be a written homework assignment almost every week. Please write up your solutions nicely, staple all the pages together, and hand them in during the following week, at the beginning of Wednesday's class. Some of the problems will be graded; the grader is Xuande Liu. We are also going to discuss some of the problems in class on the following Monday; you are expected to participate in the discussion and, from time to time, volunteer to present a solution in front of class.
| Week | Assignment |
| 1 | Problem Set 1 (due Wednesday, September 10) |
| 3 | Problem Set 2 (due Wednesday, September 17) |
| 4 | Problem Set 3 (due Wednesday, September 24) |
| 5 | Problem Set 4 (due Wednesday, October 1) |
| 6 | Problem Set 5 (due Wednesday, October 8) |
| 7 | Problem Set 6 (due Wednesday, October 22) |
| 9 | Problem Set 7 (due Wednesday, October 29) |