Math 593: Algebra I

Professor: David E Speyer
Fall 2021

The Euclidean algorithm (The Elements, Book VII)

Course meets: Monday, Wednesday and Friday, 2:00-3:00 PM, 455 Weiser Hall

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

Professor: David E Speyer, 2844 East Hall,

Course homepage:

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:

These numerical scores will be converted into letter grades typical of graduate courses.

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.

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.