Math 594: Algebra II
Professor: David E Speyer
Course meets: Monday, Wednesday and Friday, 2:00-3:00 PM, 2866
Gauss's construction of the regular heptadecagon
Office hours Mondays 3:00-5:00 PM and Tuesdays, 9:30-11:30 AM,
in East Hall 2844. I am also glad to make appointments to meet at
Professor: David E Speyer, 2844 East Hall, email@example.com
Course homepage: http://www.math.lsa.umich.edu/~speyer/594
Level: Graduate students and advanced undergraduates.
Prerequisites: Prior exposure to the definitions of groups,
rings, modules and fields, at the level of 593 or a similar course. Abstract linear algebra over an arbitrary
Structure of class: This class will be taught in an IBL style,
meaning that a large portion of the class time will be spent solving
problems that develop the theory we are studying. Students are expected to
attend class and participate in solving problems, as the class will not
work otherwise. Some portion of your grade will be allocated for
participation in class work.
Homework: I will assign weekly problem sets, due on
Exams: I plan to give two evening exams, in late February and one in
April. The problems on the exams will be very close to problems from
the class worksheets and homework; the goal is to make sure that you
are familiar with these problems and how to solve them on your
Grading: I will apportion the grade for this course as 50% problem sets and 20% from
each of the two exams, with the remaining 10% for class participation. I will drop the two
lowest problem set grades.
Extensions: I will not provide homework extensions, but please do note that I will drop
the lowest two homework grades.
Accomodations for a disability: If you think you need an
accommodation for a disability, please let me know as soon as
possible. In particular, a Verified Individualized Services and
Accommodations (VISA) form must be provided to me at least two weeks
prior to the need for an accommodation. The Services for Students with Disabilities (SSD) Office (G664 Haven Hall) issues VISA forms.
QR Exam: Many of the students in this course are preparing for
the QR exam. This course covers the material from the
Winter Term of
syllabus (and more). However, this is not a QR study course, and I would
encourage students preparing for the QR exam to take additional past
QR exams on your own. I am glad to discuss questions about those
exams, in office hours or elsewhere.
Climate: Each of you deserves to learn in an environment where
you feel safe and respected.
I want our classroom, the collaborations between my students outside
class, and our department as a whole, to be an environment where
students feel able to share their ideas, including those which are
imperfectly formed, and where we will respectfully help each other
develop our understanding. I want to provide a space where questions
are very welcome, especially on basic points.
Please ask all questions you have; remember that every question you
have is likely a question that many share. Please share your insights
and suggestions, partial or complete. Please treat your peers
questions, comments and ideas with respect.
These problem sets are written by David E Speyer and released under a Creative
Commons By-NC-SA 4.0 International License.
Homework Policy: You are welcome to consult your class notes
You are welcome to work together with your classmates provided
(1) you list all people and sources who aided you, or whom you aided
and (2) you write-up the solutions independently, in your own language.
If you seek help from mathematicians/math students outside the course,
you should be seeking general advice, not specific solutions, and must
disclose this help. I am, of course, glad to provide help!
I do not intend for you to need to consult other sources, printed or
online. If you do consult such, you should be looking for better/other
expositions of the material, not solutions to specific problems.
Math problems are often called "exercises"; note that you cannot get
stronger by watching someone else exercise!
You MAY NOT post homework problems to internet fora seeking
solutions. Although I know of cases where such fora are valuable, and
I participate in some, I feel that they have a major tendency to be
too explicit in their help. You may post questions asking for
clarifications and alternate perspectives on concepts and results we
- Problem Set 1, due Wednesday January
- Problem Set 2, due Friday January
24. The extended due date is because of Martin Luther King day
weekend; we'll return to Wednesdays after this.
- Problem Set 3, due
Wednesday, January 29. Two problems (3.2) and (3.6.2) have been
removed but will likely come back later.
- Problem Set 4, due Wednesday,
- Problem Set 5, due Wednesday,
- Problem Set 6, due Wednesday,
- Information about our first exam
- No problem set due Feb 26, in light of exam.
- 🌞 No problem set over Spring break. 🌞
- Problem Set 7, due Wednesday,
March 18. You should be able to start this immediately.
- Problem Set 8, due Wednesday,
- Problem Set 9, due Wednesday,
April 1. Students generally found problem 6, 7 and 8 hard. We
discussed 7 in class; here are solutions to 6 and 8.
- Problem Set 10, due Wednesday,
- Problem Set 11, due
Wednesday, April 15. This is the final problem set.
- Information about our second exam
These worksheets are written by David E Speyer and released under a Creative
Commons By-NC-SA 4.0 International License.
Below are the worksheets which we have used so far, and the worksheets
which I anticipate using in the next few days. Feel free to look ahead
at future worksheets before class. I do not promise to follow this
schedule, but it is my best estimate.
Here is a complete file of all the
worksheets, including two bonus sheets I didn't get to in class.
- Wednesday, January 8: The
quadratic, cubic and quartic
Status: Everyone derived the quadratic and cubic formulas!
People also made significant progress towards understanding the
quartic formula. Good first day! On Friday, we discuss quintics.
- Friday, January 10: A weak
version of the unsolvability of the quintic (slightly edited).
Status: Every group is either done or would have been done if
we had 10 more minutes. On Monday, we'll make sure everyone sees the
- Monday, January 13: Conclude Friday's worksheet. Start
Groups and Group
actions (slightly edited).
Status: All groups got through 4.4, with a few skipping
3.6. From surveys, people are happy with 4.7 and 4.8, not with 4.6
and unsure about 4.9. We'll do 4.5 and 4.6 tomorrow. I plan to skip
4.9 so here is a hint: Let H be the subgroup generated by g and look
- Wednesday, January 15: Finish 4.5 and 4.6 on Group
actions, then start Normal
groups and quotient groups.
Status: Everyone solved through 5.2, many through
5.3. Please take a look at 5.3-5.5 if you didn't get to them; these
are important. I have some misgivings, but I'm going to move on and
ask Prof. White to do Simple Groups tomorrow.
- Friday, January 17: (Professor Speyer absent, Professor White
Status: 6.1-6.4 solved. We'll need 6.5 for the future, so
we'll start with it on Wednesday.
- Monday, January 20: No class, Martin Luther King day.
- Wednesday, January 22: The Jordan
Holder Theorem (error in Problem 7.2 fixed).
Status: We started in on 6.5, but it became clear that the
worksheet was poorly designed. Putting all our ideas together, we
found two proofs anyway! We spent the last 15 minutes talking about
homework, and we'll start on 7.1 on Friday.
- Friday, January 24: More on
the Jordan-Holder theorem (error in Problem 7.2 fixed)
Status: 7.1-7.3 solved in all groups; 7.4 in active discussion
as class ended. We'll start with 7.4 on Monday.
- Monday, January 27: Finishing
the Jordan-Holder theorem
Status: 7.4 solved collaboratively, and the remaining proofs
presented as a lecture. We move to semidirect products
- Wednesday, January 29: Semidirect products.
Status: 8.1-8.3, which are the big results, proved in all
groups. 8.4 and 8.5 done in some groups and on board. Many people
requested 8.6 and 8.7 on the surveys, so we'll start with those Friday.
- Friday, January 31: Abelianization and the commutator
Status:Complete success! Prof. Weigandt tells me the class ran
very smoothly. So we'll move to solvable groups on Monday.
- Monday, February 3: Solvable
groups (minor errors fixed).
Status: 10.1-10.5 done in all groups. 10.6 done in some. We'll
do 10.6 and 10.7 briefly on Wednesday.
- Wednesday, February 5:
Nilpotent groups. Minor errors fixed in 11.3 and 11.4. Definition
of central series fixed to address an excellent criticism that I spoke
of G/Gi without requiring that Gi was normal;
Problem 11.6 added to address relation of new definition to old one.
Status: We reviewed several worksheet and homework problems
about maps of the form Ax+b, including 10.7 from last time. We then
started in on nilpotent groups. 11.1-11.3 and 11.5 solved in all groups.
- Friday, February 7: More Nilpotent groups.
Status: This material went slower than expected. 11.4, 11.6
and 11.7 solved in all groups. Prof. Speyer tried to give a big
picture discussion which didn't help. We'll try again on Monday.
- Monday, February 10: We finish Nilpotent groups.
Status: All remaining problems solved!
- Wednesday, February 12: The
Sylow Theorems (many minor typos fixed)
Status: 12.1 and 12.2 solved in all groups; 12.3 solved in
most groups; 12.4 under very active discussion at class end. We'll
pick up with 12.4 on Friday.
- Friday, February 14: More of
the Sylow theorems (many minor typos fixed)
Status: Great class! We proved everything!
- Monday, February 17: Using the
Sylow theorems (many small typos fixed)
Status: Great class! 13.1-13.8 proved. I'll leave 13.9 for you
to do on your own.
- Wednesday, February 17: Review day.
- Friday, February 21:The
Schur-Zassenhaus theorem, abelian case
Status: Great class! One group solved all problems, the other
was actively working on the last two when we left. On Monday, we'll
talk about those last two and then do the general case.
- Monday, February 24: The
Schur-Zassenhaus theorem, general case (significant edits from
Status: 15.1 was tough but both groups worked it through. 15.2
fell immediately after.
- Wednesday, February 26: Finish the Schur-Zassenhaus theorem. (significant edits from
15.3 onwward, some typos fixed)
Status: Triumph! Everyone
finished the proof and got to preview PSLn(F) is simple
(one typo fixed).
- Friday, February 28: PSLn(F) is simple
- March 2-6: 🌞 Spring break! 🌞
- Monday, March 9: Review of
Status: Widely varied
between groups. Everyone solved through 17.5, some groups finished the
worksheet. Problems 17.6-17.9 presented in a rushed fashion at the
board. We will move on, but review the later problems if your group
didn't get to them.
- Wednesday, March 11: Degrees
of field extensions and constructible numbers
Status: All problems solved
efficiently. We talked about contingency planning for the corona virus
and began Splitting fields,
solving problem 19.1.
- Friday, March 13: No class, but optional online meeting to test
our preparation for online classes. Details will be sent by e-mail on
- Monday, March 16: First online class! Splitting
fields, problems 19.2-19.5.
Status: Problems 19.2 and
19.3 solved in all groups; 19.4 and 19.5 in group 2. You might like to
look through the group-2-text channel to see if you can follow their
solutions. Next time, we will try Discord again but be logged into
Miro and see if that helps communication.
- Wednesday, March 18: Quick wrap up of previous worksheet and
start Maps between splitting
fields (20.2 improved; typo in 20.4 fixed).
Status: All problems solved,
congratulations! Remark from the future Based on discussion on
March 20, it would have been better if problem 20.4 had featured two
fields and an isomorphism between them, not an automorphism of one field.
- Friday, March 20: Field
Status: All problems solved, congratulations! The pre-corona
draft of this worksheet had some harder examples which were moved to
Problem Set 8, so you might want to work on that now.
- Monday, March 23: Galois
Status: Very nice work! Everything proved except for
22.6. We'll start with that on Wednesday.
- Wednesday, March 25: Catch up and start Separabilty, Galois closure, perfect
Status: We started by proving 22.6 and discussing some
examples. All groups then got to at least 23.2, and we did 23.3 and
23.4 together at the end. Some got as far as 23.8. We'll start with
23.5 on Friday.
- Friday, March 27: Finish Separabilty, Galois closure, perfect
Status: Nice work! Everyone zipped through these problems and
also got up to 24.2 on Artin's
- Monday, March 30: Artin's
Status: Nice work! Everything was solved by everyone, and some
groups had enough extra time to also talk about some homework
problems. I held people back from starting into the Galois
correspondence, but I expect it will go quickly on Wednesday.
- Wednesday, April 1: The
Galois correspondence (typo fixed in 25.7). This worksheet used to be two worksheets
that were then merged, and I did a poor job of it. 25.6 parts (1) and
(2) repeates 25.1 and 25.6.(3) is pretty much immediate from
definitions. Parts 25.4 and 25.5 would have been better at the end of
the worksheet. But the class did well with it anyway.
As expected, this went quickly and succesfully.
- Friday, April 3: Corollaries
of the fundamental theorem I've taken the liberty of adding a
fourth equivalent condition in 26.1.(2), which everyone wound up
proving on the way.
Status: 26.1 was challenging for everyone, but everyone got
through it. Other groups stopped somewhere between 26.2 and 26.4. We
discussed 26.2 at the end of class, so I'll ask people to pick up with
26.3 on Monday.
- Monday, April 6: More Corollaries of the fundamental theorem
Status: Complete success! Nice work!
- Wednesday, April 8: Solvable
Status: Two groups finished everything and the last went
through 27.3. We'll discuss 27.4 and 27.5 in class Friday.
- Friday, April 10: Kummer's
theorem and Galois's criterion for radical extensions (correction
in Problem 28.5). This is the
last worksheet that will be on the exam.
Status: 28.1 was difficult for a lot of groups. 28.2-4 went
pretty smoothly, some groups also started 28.5. There was also a flaw
in the statement of 28.5, which is now struck out and fixed.
- Monday, April 13, Finish Kummer's
theorem and Galois's criterion for radical extensions and
review. Please bring questions!
- Wednesday, April 15, Finish answering questions and start Symmetric polynomials and computing
Galois groups (error in 29.3 and other typos fixed)
Status: All groups solved 29.1 and 29.2, most were discussing
29.3, where they may have been harmed by an error in the worksheet.
- Friday, April 17, Finish Symmetric polynomials and computing
Status: All problems solved. One group thought I had left and
right cosets backwards in the definition of G. I think I got it right
but I haven't had a chance to check carefully, so be warned.
- Monday, April 20, Transendence bases