MAT 534: Algebra I (Fall 2025)

Announcements

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)
2 Problem Set 2 (due Wednesday, September 17)
3 Problem Set 3 (due Wednesday, September 24)
4 Problem Set 4 (due Wednesday, October 1)
5 Problem Set 5 (due Wednesday, October 8)
6 Problem Set 6 (due Wednesday, October 22)