# Math 593: Algebra I

### Fall 2019

The Euclidean algorithm (The Elements, Book VII)

Course meets: Monday, Wednesday and Friday, 2:00-3:00 PM, 4088 East Hall

Office hours Monday and Wednesday 9:30-11:30 AM, Thursday 2:00-4:00 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

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 mid-October 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.

## 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 By-NC-SA 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 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 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.

• 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, 7-9 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 By-NC-SA 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 — 7-9 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.