Math 8202: Graduate algebra

Fall semester 2025

Mon-Wed-Fri 9:05-9:55am in Vincent Hall 113
Course Canvas page, and Discord server.

Course Instructor

Victor Reiner (You can call me "Vic").
Office: Vincent Hall 256
Office hours: Mondays 4:40-5:30pm, Thursdays 4:00-5:00pm,
and on this Zoom link.
Telephone (with voice mail): (612) 625-6682
E-mail is better: reiner@umn.edu

Course Schedule

HW assignments and exam dates
Assignment,
Exam
or other event HW due date
or exam date Problems
(from Dummit and Foote
unless specified otherwise)

Homework 1 Wed, Sept. 17 From these extra HW problems:
#1,2,3,4,5,6,7,8,9,10

From Dummit & Foote
11.1 # 6,7,9
11.2 # 11
11.4 # 6

Homework 2 Wed, Oct. 1 From these extra HW problems:
#11,12,13,14,15,16,17,18,19,20,21,22,23,24
(moved Dummit and Foote 11.5 #13 to HW3)

Midterm exam 1 (in-class) Wed, Oct. 8 This was the 50 minute in-class midterm exam

Homework 3 Wed, Oct. 22 From Dummit & Foote
11.5 # 13
(and assume they meant Sym²(V) not S²(V), and Alt²(V) not Λ²(V)
7.1 # 14
7.2 # 4
7.3 # 29 (assume Exer. 25 holds),31
(removed 7.3 #4, which is boring unless you allow ring maps that don't send 1 to 1)
7.4 # 8,19,26,30,31,32
7.5 # 3
7.6 # 1,4
(but in #1, read "the subrings Re and R(1-e)" as "the rings Re and R(1-e)")
(and 7.6 #5 got moved to HW4)

Homework 4 Wed, Nov. 5 From Dummit and Foote
7.6 # 5
8.1 # 2(a),4,5(a),9
8.2 # 3,5
8.3 # 5,8(a) (removed #6,7)
9.1 # 7,13
9.2 # 1,2,4,7,8

Missed lecture Fri, Nov. 7 See Lecture notes table below for make-up video and lecture notes

Midterm Exam 2 (in-class) Wed, Nov. 12 This was the 50 minute in-class midterm exam

Homework 5 Wed, Nov. 26 10.2 # 5
10.3 # 9,10,11
12.1 # 1,2,3,4,5,6
12.2 # 3,4,11,18
12.3 # 5,16 (no need to give P with P^-1AP=J, but do justify the Jordan form J),
17 (assume A has entries in C here),22,24

Homework 6 Wed, Dec. 10 13.1 #1
13.2 #3,7,14
13.4 #2
13.5 #8
13.6 #3,4,6
14.2 #13
14.3 #3
14.5 #3,7,10
14.6 #2

Final exam (in-person)
location to be determined Mon Dec. 15
10:30am-12:30pm

Lecture notes
Date Topic

Wed Sept 3, Fri Sept 5, Mon Sept 8 Intro, definitions and examples

Wed Sept 10, Fri Sept 12 Linear algebra review

Mon Sept 15, Wed Sept 17, Fri Sept 19 Diagonalizability, triangularizability

Mon Sept 22, Wed Sept 24 Spectral theorem(s)

Fri Sept 26, Mon Sept 29, Wed Oct 1, Fri Oct 3, Mon Oct 6 Multilinear algebra

Fri Oct 10, Mon Oct 13, Wed Oct 15, Fri Oct 17 Ideals, Domains, Fields

Mon Oct 19, Wed Oct 21, Fri Oct 23, Mon Oct 27, Wed Oct 29 Euclidean domains, PIDs, UFDs

Fri Oct 31, Mon Nov 3 Polynomials and factorization

Missing lecture Fri Nov 7 Make-up video on the Ideal-Variety Correspondence, with lecture notes
(Not required course material).

Wed Nov 5, (missed Fri Nov 7), Mon Nov 10, Fri Nov 14, Mon Nov 17 Modules:Intro and examples

Wed Nov 19, Fri Nov 21, Mon Nov 24, Wed Nov 26 Noetherian Modules and PIDs

Mon Dec 1, Wed Dec 3, Fri Dec 5, Mon Dec 8, Wed Dec 10 Fields and extensions

**HW assignments and exam dates**
Assignment, Exam or other event	HW due date or exam date	Problems (from Dummit and Foote unless specified otherwise)
Homework 1	Wed, Sept. 17	From these extra HW problems: #1,2,3,4,5,6,7,8,9,10 From Dummit & Foote 11.1 # 6,7,9 11.2 # 11 11.4 # 6
Homework 2	Wed, Oct. 1	From these extra HW problems: #11,12,13,14,15,16,17,18,19,20,21,22,23,24 (moved Dummit and Foote 11.5 #13 to HW3)
Midterm exam 1 (in-class)	Wed, Oct. 8	This was the 50 minute in-class midterm exam
Homework 3	Wed, Oct. 22	From Dummit & Foote 11.5 # 13 (and assume they meant Sym²(V) not S²(V), and Alt²(V) not Λ²(V) 7.1 # 14 7.2 # 4 7.3 # 29 (assume Exer. 25 holds),31 (removed 7.3 #4, which is boring unless you allow ring maps that don't send 1 to 1) 7.4 # 8,19,26,30,31,32 7.5 # 3 7.6 # 1,4 (but in #1, read "the subrings Re and R(1-e)" as "the rings Re and R(1-e)") (and 7.6 #5 got moved to HW4)
Homework 4	Wed, Nov. 5	From Dummit and Foote 7.6 # 5 8.1 # 2(a),4,5(a),9 8.2 # 3,5 8.3 # 5,8(a) (removed #6,7) 9.1 # 7,13 9.2 # 1,2,4,7,8
Missed lecture	Fri, Nov. 7	See Lecture notes table below for make-up video and lecture notes
Midterm Exam 2 (in-class)	Wed, Nov. 12	This was the 50 minute in-class midterm exam
Homework 5	Wed, Nov. 26	10.2 # 5 10.3 # 9,10,11 12.1 # 1,2,3,4,5,6 12.2 # 3,4,11,18 12.3 # 5,16 (no need to give P with P^-1AP=J, but do justify the Jordan form J), 17 (assume A has entries in C here),22,24
Homework 6	Wed, Dec. 10	13.1 #1 13.2 #3,7,14 13.4 #2 13.5 #8 13.6 #3,4,6 14.2 #13 14.3 #3 14.5 #3,7,10 14.6 #2
Final exam (in-person) location to be determined	Mon Dec. 15 10:30am-12:30pm

**Lecture notes**
Date	Topic
Wed Sept 3, Fri Sept 5, Mon Sept 8	Intro, definitions and examples
Wed Sept 10, Fri Sept 12	Linear algebra review
Mon Sept 15, Wed Sept 17, Fri Sept 19	Diagonalizability, triangularizability
Mon Sept 22, Wed Sept 24	Spectral theorem(s)
Fri Sept 26, Mon Sept 29, Wed Oct 1, Fri Oct 3, Mon Oct 6	Multilinear algebra
Fri Oct 10, Mon Oct 13, Wed Oct 15, Fri Oct 17	Ideals, Domains, Fields
Mon Oct 19, Wed Oct 21, Fri Oct 23, Mon Oct 27, Wed Oct 29	Euclidean domains, PIDs, UFDs
Fri Oct 31, Mon Nov 3	Polynomials and factorization
Missing lecture Fri Nov 7	Make-up video on the Ideal-Variety Correspondence, with lecture notes (Not required course material).
Wed Nov 5, (missed Fri Nov 7), Mon Nov 10, Fri Nov 14, Mon Nov 17	Modules:Intro and examples
Wed Nov 19, Fri Nov 21, Mon Nov 24, Wed Nov 26	Noetherian Modules and PIDs
Mon Dec 1, Wed Dec 3, Fri Dec 5, Mon Dec 8, Wed Dec 10	Fields and extensions

Course Summary

This is a one-semester graduate core course in the algebra of

rings, fields, modules,
a bit of bilinear forms and spectral theory,
multilinear algebra (tensor, symmetric and exterior powers of vector spaces)

( We used to have a 2-semester grad algebra sequence Math 8201, 8202, roughly based on our current text, Dummit and Foote.
The old Math 8201 dealt with some of Chapters 1-6 and 11, while the old Math 8202 dealt with some of Chapters 7-10, 12-14.
As an example, see my old syllabi for 8201, 8202.)

Course Prerequisites

The prerequisites relate to the department's philosophy behind eliminating Math 8201.
Generally, we want core classes to come in 1-semester chunks, rather than the old 2-semester sequences, to give students greater scheduling flexibility.
To keep it all in one semester, the thought was to

assume that students have seen a bit of linear algebra over a field
(including notions like linear independence, spanning, bases, change-of-bases, eigenvalues, eigenvectors),
assume they have also seen a bit of group theory
(including notions like subgroups, cosets, normal subgroups, quotient groups, groups presented by generators and relations, examples such as permutation groups and matrix groups),
focus more on the rings, fields, and modules, multilinear algebra
de-emphasize some proofs, since there are now more tools and resources available explaining proofs.

Students who have taken a proof-based undergrad "groups, rings, fields" class should be fine. This includes our own Math 5285-5286 sequence.
However, those who have taken only our Math 4281 may need extensive self-study to prepare for the class.

Here are some notes on dihedral groups intended to illustrate some of the terminologies, definitions, concepts, notation that we will be assuming is familiar.
For things in those notes that look unfamiliar, consult the instructor.

Course Goals and Objectives

By the end of this course, we would like to students to gain familiarity and facility with the following topics,
and may even end up discussing them in this order:

commuting operatrs and simultaneous diagonalizability, triangularizability
bilinear and quadratic forms,
(finite-dimensional) spectral theorem for self-adjoint and normal operators,
multilinear algebra, including tensor products, symmetric and exterior products (over fields)
rings, ideals, and modules, commutative and noncommutative,
Jordan-Hölder theorem
principal ideal domains, Smith normal form, and structure theorem for modules
Jordan and rational canonical forms
unique factorization domains
Noetherian rings and modules, Hilbert basis theorem
field theory, including Galois theory

Course Format

We mainly use a lecture format. There may be occasional active learning exercises.

Course links

Our course Canvas page.
Our class Discord server. Using it to discuss homework and form study groups are strongly encouraged.
Our text: Abstract Algebra by Dummit and Foote, 3rd edition, Wiley 2004.
Other useful sources
- Paolo Aluffi's Algebra: Chapter 0
- Mike Artin's Algebra
- Robert Ash's Abstract algebra: the basic graduate year
- John Beachy's Abstract algebra online
- Keith Conrad's expository blurbs
- Paul Garrett's Abstract Algebra page
- Tim Gowers's How to lose your fear of tensor products
- Thomas Hungerford's Algebra
- Serge Lang's Algebra
- Patrick Morandi's Field and Galois theory
- Joseph Rotman's Advanced modern algebra
- Ian Stewart's Galois theory
- The department's old algebra written prelim exams
- My old Math 8202 syllabi from 2002, 2011, 2014, 2020.

Textbook and Required Materials

Only the previously mentioned Abstract Algebra by Dummit and Foote.

Course Grading

The course grade is based on a combination of homework and exams.
There are 6 homeworks due every two weeks on Wednesday evenings around midnight,
two in-class (50 minute) midterm exams and an in-person final exam during finals week. All homework is to be turned in at the course Canvas site.
See the schedule table above for the assignments and dates.

Exams

Midterm exam 1: in-class on Wed. Oct. 8
Midterm exam 2: in-class on Wed. Nov. 12
Final exam: in-person, 10:30am-12:30pm on Mon. Dec. 15

Grading

Homework = 40% of grade
Midterm 1 = 20% of grade
Midterm 2 = 20% of grade
Final exam = 20% of grade

Course Policies

I encourage collaboration on the homework, as long as each person understands the solutions and writes them up in their own words. Indicate on the homework page any collaborators, and/or resources used, including AI tools such as ChatGPT.
For homework it fine to use online sources including AI tools. But be aware that, if you haven't fully absorbed (and checked) what they are telling you, then you may find yourself at a disadvantage during midterm and final exams where no electronic devices will be allowed.
As mentioned above, unlike the homework, during in-class exams, no outside sources (electronic or otherwise) are to be used, beyond one 3 x 5 index card of notes that you may write on in advance, both sides.
Homework solutions should be well-explained-- the grader is told not to give credit for an unsupported answer.
Complaints about the grading that are not easily resolved with the grader should be brought to me.
We will only be using the course Canvas site for turning in the homeworks as PDFs.
If you write solutions by hand, then use a scanning app (e.g., Adobe Photo Scan) or a scanner to create the PDFs.
Do not just take a photo and convert it to PDF, as those are hard to read.

Attendance

I will not be checking attendance, but attending all lectures is strongly encouraged.
There is a significant amount of material that may appear in lectures,
which is not in Dummit and Foote, but gleaned from some of the other sources in this syllabus.

Make-up/Dropped Exams and Late Work

Late homework will not be accepted, without accommodations from Disability Servcies
or medical excuses written by a medical professional.
There will be no make-up or dropped exams or homeworks.

Incompletes

The grade I ("incomplete") shall be assigned at the lecturer's discretion when, due to extraordinary circumstances,
the student was prevented from completing the entire course. It is my policy to assign incompletes only rarely,
and only when almost all of the course has already been completed in a satisfactory fashion prior to the extraordinary circumstances.
See me if something occurs which makes you think you should receive an incomplete.

Required University Policy Statements

Education & Student Life Policies
Student Conduct Code
Use of Personal Electronic Devices in the Classroom
Scholastic Dishonesty
Excused Absences and Makeup Work
Appropriate Student Use of Class Notes and Course Materials
University Grading Scales
Sexual Harassment, Sexual Assault, Stalking and Relationship Violence
Equity, Diversity, Equal Opportunity, and Affirmative Action
Disability Accommodations
Mental Health and Stress Management