Course progress

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.

• August 26, Rings Download Rings
  • Status: Problems 1.1-1.6 solved nicely in class. You can think about 1.7 on your own (and I'm glad to discuss it with you).
• August 28, Modules Download Modules
  • Status: 2.1-2.4 solved in all groups, 2.5 in some groups, and 2.4-2.5 discussed on board. This was a bit rushed, and I'm glad to discuss it more slowly with anyone who would like, but we'll move to ideals on Friday.
• August 30, Ideals Download Ideals
  • Status: All problems were solved quickly; I should have brought more! So we started talking about material from the next worksheet and solved problems 4.1, 4.2 and 4.5. We'll pick up with the rest of worksheet 4 next time, and possibly Worksheet 5 as well.
• September 2, Labour Day, no class
• September 4, Prime and Maximal Ideals Download Prime and Maximal Ideals
  • Status: All problems were solved quickly and straightforwardly.
• September 6, Comaximal ideals Download Comaximal ideals
  • Status: Problems 5.1-5.5 solved in all groups, despite some cleverness being required. Problems 5.6 and 5.7 solved in many, and we even had time for a digression about the Nullstellensatz.
• September 9, Product Rings, Product Modules, and the Chinese Remainder Theorem Download Product Rings, Product Modules, and the Chinese Remainder Theorem
  • Status: All problems were solved quickly. On the homework, we'll see some applications. In class, we'll move on toward the Jordan-Holder Theorem.
• September 11 and 13: We paused worksheets for a while to discuss examples of modules and computations with direct sums.
• September 16, Simple modules Download Simple modules
  • Status: All problems solved.
• September 18, Composition series Download Composition series
  • Status: All problems discussed, but not all discussed in class. We'll come back to 8.7 briefly next time.
• September 20 and 23, The Jordan-Holder theorem Download The Jordan-Holder theorem
  • Status Sept 20: First four problems solved well in class.
  • Status Sept 23: All problems solved in all groups
• September 25, Finitely Generated Modules Download Finitely Generated Modules
  • Status Sept 25: Problems 10.1-10.4 solved well and discussed in class. We'll skip over problem 10.5, and discuss 10.6 and 10.7 briefly on Friday.
• September 27 Noetherian rings Download Noetherian rings
  • Status: All problems solved, although the 1(a) ⇒ 1(b) ⇒ 1(c) implications were rushed at the end. That's okay, they'll be on the problem set!
• September 30 The Krull-Schmidt theorem (lecture)
  • Status: Well, I had fun! I hope that the proof worked for all of you. I really like it as a capstone to the various tools that we've learned.
• October 2 and 4, Unique Factorization Domains Download Unique Factorization Domains
  • Status October 2: (Professor Snowden replaces Professor Speyer): Good work, first six problems solved.
  • Status October 4: (Professor DeBacker replaces Professor Speyer): All problems completed.
• October 7, Principal Ideal Domains Download Principal Ideal Domains
  • Status: The standard results 13.1-13.8 were proved in all groups. The fun result 13.9 was proved in some groups, and was presented at the board on Oct 9.
• October 9 and 11, The Euclidean Algorithm Download The Euclidean Algorithm
  • Status October 9: Problems 14.1-14.4 were proved in all groups. We'll talk about 14.5, and assorted fun topics, on Oct 11. We may or may not get to starting the next worksheet.
  • Status October 11: We talked about some interesting properties of 2x2 matrices and how they related to the Euclidean algorithm. At the end, we defined a Euclidean doman.
• October 14: 🍂 🍁 🍂 🍁 🍂 Fall break.  🍂 🍁 🍂 🍁 🍂
• October 16 and 18, Euclidean Domains Download Euclidean Domains
  • Status October 16: We solved all the most important problems: 15.1-15.5. We are doing well on time, though, so we'll finish the remaining problems on Friday.
• October 21: (Professor Chan replaces Professor Speyer) Introduction to Smith Normal Form Download Introduction to Smith Normal Form
  • Status: All problems solved except for proving the Cauchy-Binet theorem; we'll postpone that for another day.
• October 23: (Professor Chan replaces Professor Speyer) Proof of the Smith Normal Form Theorem Download Proof of the Smith Normal Form Theorem
  • Status: All problems solved; some though not all problems presented on the board. We'll move on. If there is a problem that you in particular didn't see the solution to, please feel free to ask a classmate or Prof. Speyer.
• October 25, 28, 30: Classification of finitely generated modules over a PID Download Classification of finitely generated modules over a PID
  • Status October 25: Problems 18.1 and 18.2 solved and discussed on the board; some groups got further. We'll pick up here on Monday.
  • Status October 28: Problems 18.3-18.5 solved in all groups. We will finish this on Wednesday.
  • Status October 30 Finished, in lecture form.
• November 1 and 4: Rational canonical form Download Rational canonical form.
  • Status November 1: Problems 19.1-19.4 largely finished in groups
  • Status November 4: Problems 19.5 and 19.6 finished in class discussion
• November 6: Jordan and generalized Jordan form Download Jordan and generalized Jordan form
  • Status: All problems solved.
• November 8: The class semigroup and rings that are "close" to Euclidean (lecture)
• November 11, 13 and 15: Unique factorization in polynomial rings Download Unique factorization in polynomial rings. One problem (problem 6) added on November 13.
  
  • Status November 11: Problems 1, 2 and 3 solved in class. Problem 4 solved in most groups. We'll pick up with Problem 4 on Wednesday.
  • Status November 13: All problems solved, but no time for discussion. We'll have the discussion on Friday.
  • Status: November 15: We talked through all the proofs, but it took longer than expected.
• November 18: Tensor products of vector spaces Download Tensor products of vector spaces
  • Status November 18: Problems 1-4 done in class. We'll do Problem 5 as a lecture on Nov. 20, to make sure that we stay on schedule.
  • Status November 20: Problem 5 proved in lecture
• November 20 and 22: Symmetric and exterior algebras Download Symmetric and exterior algebras
  • Status November 20: All the relevant definitions presented in lecture, but we didn't get to solving problems. That has me nervous; I hope we'll catch up on Friday.
  • Status November 22: Problems on the symmetric algebra (23.1-23.5) close enough to solved that I am willing to call them done. Problems on exterior algebra still largely untouched; we'll finish this on Monday.
• November 25: To be determined.
• December 2: Bilinear forms Download Bilinear forms
  • Status December 2: All problems except the last one solved in groups.
  • Status December 4: Last problem solved in classwide discussion.
• December 4-6: Symmetric bilinear forms Download Symmetric bilinear forms
  • Status: December 4: First three problems solved.
• December 6: Real symmetric bilinear forms Download Real symmetric bilinear forms
• December 9: To be determined.