Math 593: Algebra I
Professor: David E Speyer
Fall 2019
Course meets: Monday, Wednesday and Friday, 2:003:00 PM, 4088
East Hall
Office hours Monday and Wednesday 9:3011:30 AM, Thursday
2:004:00 PM, 2844 East Hall. I am also glad to make appointments to
meet at other times. At the moment, I am making all my office hours
open to both my classes (593 and 665); if this causes a problem, I may restructure.
Professor: David E Speyer, 2844 East Hall, speyer@umich.edu
Course homepage: http://www.math.lsa.umich.edu/~speyer/593
Level: Graduate students and advanced undergraduates.
Prerequisites: Prior exposure to the definitions of groups,
rings, modules and fields. Abstract linear algebra over an arbitrary
field.
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. I am indebted to
Stephen DeBacker for writing problem sheets to make this possible when
he taught the class in Fall 2018; I have extensively modified these
problem sheets for the upcoming term. 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
Fridays. These problem sets will be more substantive than a typical
graduate course, but they will not be the longest ones ever assigned
in Math 593.
Exams: I plan to give two in class exams, in midOctober and one in
December. 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
own.
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. The number thus obtained will be converted to a letter grade
by a fairly generous curve.
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 Fall term of
the QR
syllabus (and more). Several problems from past QR exams will appear on
problem sets; my current drafts have at least one every week. That said, 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.
Problem Sets
These problem sets are based in part on earlier problem sets by
Stephen DeBacker, (c) 2018 UM Math department, under a Creative
Commons ByNCSA 4.0 International License. I release these under
the same license.
Homework Policy: You are welcome to consult your class notes
and textbook.
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 writeup 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
have covered.
Homework formatting: In order to make our grading process
efficient, please write on only one side of the page and place your
problems in order. If your solution to a problem is lengthy (more than
2/3 of a page say), please don't write solutions to other problems on
that page.
Also, please mark the homework with your UMID number rather than your name.
 Problem Set 1, due Friday September 13
 Problem Set 2, due Friday September
20
 Problem Set 3, due Friday September
27
 Problem Set 4, due Friday October
4. Note: Problems 36(d) and (e) are extended to Problem Set 5
(which will be due October 11.)
 Problem Set 5, due Friday October
11.
 🍂 🍁 No problem set due on October 18. Enjoy Fall break! 🍂 🍁
 Problem Set 6, due Friday October
25. Note: This problem set contains information about the
exam.
 Exam on Wednesday October 30, no problem set due on Friday,
November 1.
 Problem Set 7, due Friday November 8.
 Problem Set 8, due Friday November
15.
 Problem Set 9, due Friday November
22.
 Our second exam is Thursday, December 5, 79 PM, in East
Hall 3088. Here is a study guide for the exam.
 Problem set 10. This is our last problem
set! In view of
Thanksgiving and the exam, it will be due on Monday, December 9.
Class worksheets
These worksheets are based in part on earlier problem sets by
Stephen DeBacker, (c) 2018 UM Math department, under a Creative
Commons ByNCSA 4.0 International License. I release these under
the same 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. Especially in the early days of
class, I expect that I will have to make a lot of revisions, so don't
count on me following the announced schedule.

Wednesday, September 4: Rings.
Status: (1)(7)
done. We'll discuss (8) and (9) next time and leave (10) as a fun topic
for people to consider on their own.
 Friday, September 6: Modules.
Status: Problems (8), (9) and (11)(14) done; some groups started (15). We'll start with (15) next time.
 Monday, September 9: Modules again,
Ideals.
Status: (15)(17) solved in all groups, (18)(20) in almost all. I'll say a few words about (17) and (18) at the start of next
class, and then we'll start with (21).
 Wednesday, September 11:
Ideals again (slightly
edited).
Status: (22)(25) solved in all groups. (21) started again. We'll
spend some time on (21) next time, but mostly move on.
 Friday September 13:
Prime and Maximal
Ideals (this version slightly edited from class
version).
Status: (21) and (26)(29) solved in all groups; (30) being actively discussed at end of class.
We'll pick up with (30) next time and leave (31) and (32) for you to think through on your own.
 Monday September 16:
Comaximal ideals.
Status: (30) and (33)(35) solved in class. (36) and (37) in
progress. We'll pick up with (36) and (37); we may skip (38)(40)
depending on time.
 Wednesday, September 18:
Products of Rings. We'll also
spend some time talking about the first Problem Set.
Status: Problems (36) and (37) done in class; problems (41), (42) solved in all groups. Problem (43)
was well under way when class ended. Some groups also went back and solved
part or all of (38)(40).
 Friday, September 20: The
Chinese Remainder Theorem.
Status: All problems solved in all groups. In addition, many groups
went back and finished (43) and/or (38)(40). Nice work!
 Monday September 23:
Noetherian Rings (missing word
"nonempty" added in third column)
Status: (1) → (2) → (3) → (1)
completed. (a) ← (b) ↔ (c) completed. (a) → (b)
under active discussion at end of class.
 Wednesday, September 25:
Finished up Noetherianity, then started Unique Factorization Domains
Status: (50)(51) solved in all groups. (52)(54) in progress
as we ended.
 Friday, September 27:
Unique Factorization Domains
again. (This is the version given in class, acrhived here for
historic reasons. See the following day for a better version.)
Status: Problems (52)(56) solved in all groups. Good
discussion of problem (57), but discussion made me realize the
worksheet was poorly designed on this point. Expect a better version
on Monday.
 Monday, September 30: Rosh Hoshanah, Professor Smith subsitutes
for Professor Speyer. Unique Factorization
Domains (edited). The old (57) has been replaced by new (57) and
(58). Start at the new version of (57) and go from there.
Status: Problems (58)(60) solved in all groups. Problem (57)
remained hard, but was solved in some groups. We'll discuss (57), and
also summarize our results as a whole, next time.
 Wednesday, October 2: The Euclidean
Algorithm. I'm moving this material before PID's to make sure you
have it in time to solve these problems on the problem set.
Status: Problems (61)(64) solved in all groups, problems
(65)(66) solved in most groups and a few minutes away from solution
in the rest. We won't
spend more time on this worksheet, but please feel free to ask Prof. Speyer
questions if you have doubts.
 Friday, October 4: Euclidean rings. Slightly
edited from version in class.
Status: Problems (67)(71) solved in all groups, which are the
imortant problems. Most groups got at least (72) and some got as far
as (74); good work! We'll move to PID's on Monday.
 Monday, October 7: Principal Ideal Domains.
Status: All problems solved with lots of time to spare, so we
also did this summary worksheet. Almost all groups got to the end of
the summary worksheet and we discussed it together. Nice work!
 Wednesday, October 9: Yom Kippur, Professor DeBacker
substitutes for me. We finally get to Products from a Categorical
Perspective.
Status: (87)(90) solved in all groups, (91) under discussion
at end of class.
 Friday, October 11: Bonus fun topic. Come to class and see
what happens!
 Monday, October 14: 🍂 🍁 FALL BREAK!
🍂 🍁
 Wednesday, October 16: Professor Speyer away, Professor
DeBacker substitutes. Introduction to Smith Normal Form.
Status: (92)(97) solved in all groups, (98) started in
some. Nice work! We'll leave (98) as something to return to later and
move on next time.
 Friday, October 18: Proof of the Smith Normal Form
Theorem.
Status: Two complete proofs of the Smith Normal Form Theorem
were given, with complete solutions to (99), (101) and (102). Problem
(100) was solved in one group; I'll be glad to help people thinking
about that on their own, but we will move on.
 Monday, October 21: Classification of finitely generated modules over a PID
(errors in (103) corrected)
Status: (103)(105) solved in all groups and (106) in all but
one. We'll quickly do (106) on the board on Wednesday, and then pick
up with (107).
 Wednesday, October 23: Classification of finitely generated
modules over a PID again.
Status: I had forgotten how hard uniqueness is! People made
valiant efforts at (107), but it fought back! Some groups decided to go ahead and do (108) and (109); everyone
who tried this did it fine. (108) and (109) are good problems to look at on your
own if you haven't yet.
 Friday, October 25: We finish uniqueness (problem
107). Rather than introducing a new worksheet after this, I'll put
aside time to answer questions from homework and old
worksheets. Please come prepared with questions!
 Monday, October 28: More review.
 Wednesday, October 30: Rational canonical form. Note Exam in evening.
Status: (110)(113) solved in all groups; some groups also
solved (114) and/or (115). We'll move on; problems (114) and (115) are
likely to reappear on problem sets in the future.
 Friday, November 1: Jordan Normal Form. Some of today's
class will be reserved for a focus group concerning the structure of
the class, conducted by a master's student from the School of Education.
Status: (117) solved in all groups, (118) done in most and
almost done in all, (119) done in some. We'll pick up with (119) on Monday.
 Monday, November 4: Finish Jordan Normal Form and do Computing Jordan normal form.
Status: (118) and (119) finished on the board. (121)(123)
solved in all groups and (124) in almost all. We'll leave the
remaining problems for you to do on your own and move on.
 Wednesday, November 6: Unique factorization in polynomial
rings (slightly edited).
Status: A productive class! (127)(132) finished in all
groups; the climax problems (133) and (134) were in progress as we
ended. We'll finish them on Friday.
 Friday, November 8: Finish Unique factorization in polynomial
rings and catch up.
Status: Complete triumph! We went through the proofs of
(127)(134) as a group and even had time to discuss a difficult
homework problem.
 Monday, November 11: Tensor products of vector spaces
(typos in (138) fixed).
Status: Most groups did (135)(143) and either finished (144)
or were working on it when we stopped. One group was new to defining
vector spaces by generators and relations and went a lot slower. I am
going to move to the next worksheet, but I would encourage people who
didn't get to do (139)(143) to think about them on their own and
bring questions to office hours if they need help.
 Wednesday, November 13: Tensor products of modules (typos
fixed in (147) and (149))
Status: (145)(150) solved in all groups, (151) in active
progress at the end of class. We'll pick up wih (151) next time. Also,
if you didn't use commutativity in your solution to (148), you are
missing something; think about what.
 Friday, November 15: Finish Tensor products of modules and
then More tensor product problems
Status: (151) presented on the board. (152)(156) solved in all
groups. (157) and (158) presented on the board. We'll move on for
now. (161) will appear again either on a worksheet or problem set,
but I am not satisfied with the route I provided, so I'll see if I
can simplify this problem before it comes back.
 Monday, November 18: Tensor products of rings
Status: (162)(164) solved in all groups, various amounts of
progress made beyond that. We'll discuss (165) and (167) on Wednesday;
and possibly (168). Note: Due to a
version control error, the problem numbering skipped from 161 to 163
on the class handouts. The numbers have now been corrected in this web
version, and the status update refers to the corrected numbers.
 Wednesday, November 20: More Tensor products of rings and Tensor, symmetric and exterior
algebras
Status: (165) and (167) done on board. (169)(171) done in all
groups and (172) mostly done. I'll say a few words about (172) next
time and then restart you at (173).
 Friday, November 22: More Tensor, symmetric and exterior
algebras
Status: Varied. Some groups moved through most of the exterior
algebra material (174)(176) and started in on (177) or (178), some
just got to the end of (173).
My tentative plan is to try to cover up through (178) as a lecture and
start you on (179) on Monday.
 Monday, November 25: Finish Tensor, symmetric and exterior
algebras
Status: (173)(177) done on board; all groups did (179)(181),
(182) under active discussion at end of class. We'll do (182) quickly
next time and then move on.
 Wednesday, November 27: Bilinear Forms (Finite
dimensionality hypothesis added to (188) and (189). In class, I
thought that (189) was missing a symmetry hypothesis, but the problem
was correct, although possibly misleading, as written. I added a note to the worksheet explaining the
issue.)
Status: Attendance was sparse. Students who were there solved
(183)(188) and were working on (189) when class ended. If you weren't
in class, please look at this worksheet before the next class!
 Friday, November 29: THANKSGIVING BREAK!
 Monday, December 2: Symmetric Bilinear Forms
Status: Problems (183)(189) were presented on the
board. Problems (190) and (191) were solved in all groups. Everyone
understood how to do the computation in (192) but not everyone finished
the computation in (192b).
There is more than one correct answer, but one answer is to take the basis (1,0,0), (1,2,0),
(1,2,3) and the diagonal matrix to be diag(2,6,12). Different groups
did different subsets of (193)(195). On Wednesday we'll take a break
and answer questions on old material, then return quickly to
(193)(195) Friday.
 Wednesday, December 4: Catch up/Review. Please bring questions!
 Thursday, December 5: Exam in evening — 79 PM, East Hall
3088
 Friday, December 6: Real Symmetric Bilinear Forms
(error in (199) fixed; pizza joke deleted since it caused more confusion than
smiles)
Status: (193)(195) presented on board; (196)(198) solved
in all groups. (199) had a confusing error, which is corrected in the
current website version. Next time, we will look at (199)(201).
 Monday, December 9: Finish Real Symmetric Bilinear Forms (another error fixed in 199) and
start Tensors in Physics.
Status: (196) and (199) presented at board; another error was
found in (199)! My apologies, it should be right now. (200) and (201)
solved in all groups, and (202) in some. (203) and (204) solved in all
groups and (205) in some.
 Wednesday, December 11: Last day of class! We'll turn over
the physics worksheet Tensors in
Physics and look at the problems on quantum mechanics. This will
let us end the class with Bell's theorem, which is mathematically not
that deep, yet extremely disturbing.