Math 593: Algebra I

The Euclidean algorithm (The Elements, Book VII)

Office hours:Tuesday and Friday, 12:00 PM-1:30 PM, 2844 East Hall.

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.

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.

Finding a generator of an ideal in the integers, Aryabhatiya

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.

Gradescope: This course will use Gradescope to handle homework and weekly quizzes. The Gradescope page is here. Please ask for an access code for the course if I haven't given one already.

Homework: I will assign weekly problem sets, due on Wednesdays. Each problem set will include a requirement to write up solutions to problems from class. See below for homework policies.

Gauss's Lemma, Disquisitiones Arithmeticae

QR practice: Most (perhaps all) students in this class are preparing to take the QR exam in algebra. This course will cover the vast bulk of the material from the Algebra 1 syllabus (and more).

I suspect that, for many students, the problem with QR exams may not be knowing the relevant material, but practicing taking the exams. Therefore, each week, I will assign a timed quiz on Gradescope consisting of two QR questions, on topics related to the current class material, to be done within a one hour period. See below for quiz policies.

Grading: In my ideal world, this class would not have grades. Almost all of you are taking the QR exam, and almost all of you have already gotten into graduate school, so it isn't clear to me what the point of grades would be. However, I am required to state a grading policy, so here one is: A numerical score will be computed as follows:

• 50% homeworks not including the write ups of class problems, with the lowest two weeks dropped
• 30% write ups of class problems, with the lowest two weeks dropped
• 10% QR practice quizzes, with the lowest six weeks dropped
• 10% Class participation

Class worksheets:

This section of the website will record worksheets which were used in class, our progress on them, and my anticipated future worksheets. Worksheets posted for future dates are subject to change.

Here are all the worksheets, plus a number of unused ones, in a single PDF file.

• August 30: Rings
• Aug. 30: Problems 1.1-1.6 solved. We'll discuss Problem 1.8 at the start of next class and leave 1.7 and 1.9 for you to think about.
• September 1-3: Modules
• Sept. 1: Problems 2.1 and 2.2 solved in all groups; various levels of progress on remaining problems. We'll talk about direct sums next time.
• Sept. 3: We discussed direct sums in the first half of class. Here is the nice categorical lemma I mentioned: If M₁₂, ..., M_r and N are modules, and we have maps e_i:M_i → N and f_i: N → M_i obeying f_i e__i = 1, f_i e_j=0 for i ≠ j and Σ e_if_i = 1, then N is the direct sum of the M_i. I think this is a nice exercise for anyone learning to think categorically.
• September 3: Ideals
• Sept. 3: All problems solved.
• September 8: Integral Domains
• Sept. 8: Problems 4.1-4.6 solved in all groups. The construction of the field of fractions (Problems 4.7 and 4.8) has a lot of details, and most people didn't finish checking them all, but we'll leave that for you to finish off at home.
• September 10: Prime and Maximal ideals
• Sept. 10: Problems 5.1-5.5 solved in all groups. The remaining problems are mostly about examples. They are good examples, but we'll move on.
• September 13: Products of rings
• Sept. 13: A good class! All problems were solved and we had a number of interesting discussions about what idempotents are good for, as well as leaving some topics to discuss next time.
• September 15: Comaximal ideals
• Sept. 15: The problems which will be key for next time — 7.1-7.5 — were all solved. If we have some extra time, we'll look at 7.6-7.8 on Sept. 17.
• Sept. 17: We discussed 7.6 and 7.7.
• September 17: The Chinese Remainder Theorem
• Sept. 17: Great class! We proved everything and had time to talk for a bit about the Nullstellansatz at the end of class.
• September 20: Simple modules
• Sept. 20: All theorems proved, and we had a chance to preview the Jordan-Holder theorem.
• September 22-24: Composition series
• Sept. 22: Problems 10.1-10.4 were solved. There was good discussion of 10.5 but we didn't resolve it, so we'll return to this issue on Friday. Many students noticed that the hypothesis that k was central in 10.3 wasn't needed; it has been removed.
• Sept. 24: We finished the problems!
• September 27-29: The Jordan-Holder theorem
• Sept. 27: A long class discussion proved 11.1
• Sept. 29: The theorem is proved! Good class. Some students pointed out that it would be better if 11.1 explicitly stated that, when A_i ≠ A_i-1, then A_i/A_i-1 ≅ B_i/B_i-1; this has been added.
• October 1: Direct sums and idempotents (lecture).
• October 4-6: Noetherian Rings.
• Oct. 4: Great class! We proved (1) ⇒ (2) ⇒ (3) ⇒ (1) in every row, (c) ⇒ (b) ⇒ (a) in every columnn, and 1(b) ⇒ 1(c). What remains is to get from row (a) to row (b)! Minor improvements to worksheet: "R" corrected to "r" in column 2; word "properly" added to column 3 for clarity.
• Oct. 6: We proved the final implication; it was a nice team effort. We also had time to talk generally about the difference between "finite length" and "finitely generated", and the difference between "relatively prime" and "comaximal".
• October 8-13: Unique Factorization Domains
• Oct. 8: Good class. We discussed 13.1-13.5, and I think 13.6 was also solved by everyone; we'll finish 13.7 and 13.8 on Monday.
• Oct. 11: We finished 13.7 and 13.8, up to a missing set theoretic result: König's Lemma. I'll address this on Wednesday.
• Oct. 13: We finished the discussion of 13.8, both explicitly using Konig's Lemma and showing how to sidestep the issue (at the cost of using definition 1(c) of Noetherian instead of 1(b)).
• October 13: Principal Ideal Domains
• Oct. 13: Problems 14.1-14.8, problem 14.9 and 14.10 punted to homework. Some groups found it more natural to first show that, in a PID, irreducible elements generate maximal ideals, thus knocking off 14.5 and 14.8, and then come back to do 14.3, 14.4, 14.6 and 14.7 as corollaries of this. I like this approach; I might rewrite the worksheet to use it in a future term.
• October 15: The Krull-Schmidt theorem (lecture).
• October 18: 🍁 🍂 🍃 Fall Break 🍁 🍂 🍃
• October 20: The Euclidean Algorithm
• Oct. 20: All problems more or less solved, modulo people not actually wanting to multiply matrices. Note that several of the solutions can be simpler if the goal is just to develop the Euclidean algorithm and not, simultaneously, to make the connection to the concept of elementary matrices.
• October 22-25: Euclidean Domains
• Oct 22: The key problems, 16.1-16.5 were all solved. We'll dicuss the fun ones, and maybe some more fun results, on Monday.
• Oct. 25: We saw how our results had concrete consequences in number theory, like "if -1 is square modulo a prime p then p=a²+b²", "if -2 is square modulo a prime p then p = a²+2 b²" and "every positive integer is a sum of four squares". The last one was a bit rushed, but I hope people still got to see how the Euclidean condition paid off.
• Last time, I finished this portion of the course with this worksheet.
• October 27-29: Introduction to Smith normal form Typo corrected: |I|=|K|=q, not |I|=|J|=q in the statement of Cauchy-Binet.
• Oct. 27: We worked through 17.1 and 17.2 and learned how to think about change of basis and isomorphism.
• Oct. 29: We did 17.3-17.6, and we now know how to compute elementary divisors.
• November 1-3: Proof of the Smith normal form theorem
• Nov. 1: Great class! Everything was proved, but we had no time for wrap up. We'll wrap up on Wednesday.
• Nov. 3: We went through the proofs.
• November 5-10: Classification of modules over a PID
• Nov. 5: We did 19.1-19.3 carefully, and 19.4 quickly at the end, which means we have proved that every finitely generated R-module is isomorphic to one of the ones in our list. We haven't proved that these modules are non-isomorphic yet, that will be Monday's project.
• Nov. 8: We finished the main computation (parts 1 and 2 of 19.5) but not the application to finish the proof. We'll finish this at the start of next class.
• Nov. 10: Finally done! We showed that all the distinct rational canonical forms are different.
• November 10: Rational normal form (lecture). If we had done this class as IBL, we would have used this worksheet.
• November 12: Jordan normal form (lecture). If we had done this class as IBL, we would have used this worksheet.
• November 15-19: Using Jordan normal form
• Nov. 15: We got most of the way through 20.1 and 20.2 and finished them on the board. We'll return to this on Wednesday.
• Nov. 17: We did 20.3 on the board and got most of the way through 20.4 and 20.5 in our groups.
• Nov. 19: We had a really nice presentation of a proof that all real symmetric matrices are diagonalizable, and we saw a lot of the results on regular matrices. I thought after class about how to present the regular matrix material better, and I decided the answer was to generalize it: See Problem 4 on Problem Set 10.
• November 22: Tensors (lecture). Here is the worksheet I used when I taught this last time.
• November 24 and 26: Thanksgiving break
• November 29-December 1: Unique factorization in polynomial rings.
• Nov. 29: Problems 21.1-5 solved, although the solution to 21.5 had to be presented in a bit of a rushed way.
• Dec 1: All problems solved!
• December 3: Exterior algebras (lecture). Here is the worksheet I used when I taught this last time.
• December 6: Some problems about exterior algebra.
  • Dec. 6: Problems 22.1-22.4 solved; we'll skip over the rest.
• December 8: Symmetric bilinear forms (lecture). Here is a worksheet on bilinear forms and one on symmetric bilinear forms from the last time I taught this..
• Decemeber 10: Signature of real symmetric bilinear forms (lecture). Here is a worksheet on signature from the last time I taught this.

Weekly assignments

Homework PoliciesYou 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.

Quiz policies Just as on the QR exams, please schedule a single uninterrupted time period to take this quiz and please complete the quiz without aid of any other resources, including written notes, internet references or other people.

I hope and believe that this practice will be useful beyond the QR exam. I think that the ability to solve problems which take 5-20 minutes is what unlocks the ability to solve problems that take months or years. I should say that this is something where different mathematicians experience varies wildly: I have found my ability to prove and disprove minor claims quickly has been extremely helpful in letting me explore difficult areas without getting lost; other mathematicians whom I greatly respect disagree. I hope that giving you some practice in this skill will be at least of some help.

I also encourage students to attempt other past QR exams. I am glad to discuss problems on these exams with you.

Assignments Due to Memorial Day and Rosh Hoshanah, the first problem set and first quiz will be due on Wednesday, September 15.

• Problem Set 1 (TeX), due Wednesday, September 15.
• Problem Set 2 (TeX), due Wednesday, September 22.
• Problem Set 3 (TeX), due Wednesday, September 29. Note: Problems first two parts of Problem 5 removed because they were done in class. Problem 9 postponed to Problem Set 4.
• Problem Set 4 (TeX), due Wednesday, October 6.
• Problem Set 5 (TeX), due Wednesday, October 13.
• There will be no problem set or quiz due on October 20. Enjoy your break! The next problem set will be due October 27.
• Problem Set 6 (TeX), due Wednesday, October 27.
• Problem Set 7 (TeX), due Wednesday, November 3.
• Problem Set 8 (TeX), due Wednesday, November 10.
• Problem Set 9 (TeX), due Wednesday, November 17.
• There will be no problem set or quiz due on November 24. There will be a problem set and quiz due on December 1 3, as well as December 8.
• Problem Set 10 (TeX), due Friday, December 3. Apologies for getting this posted late; in compensation, I have extended it until Friday, December 3.
• Problem Set 11 (TeX), due Wednesday, December 8.