MATH 681 Spring 2021
Basic Information
- Instructor: Jeff Thunder
- Office: WH 32 (and sometimes 362), but since the class is online, this doesn't matter
- Phone: 753-6725 (or use the main office phone number)
- e-mail: jthunder@niu.edu
- Office Hours (virtual): Monday, Wednesday, Friday, 12:00-12:50 p.m. or by appointment.
Text and Syllabus
The textbook for the course is Number Fields by Daniel A. Marcus. We will attempt to cover as much of the material from the first seven chapters as is reasonable.
Course Description and Objectives
This is a graduate course in algebraic number theory. The emphasis will be on algebraic number fields and their respective rings of integers. We will begin the course by defining those words, considering some examples, and investigating a few important structural properties thereof. We'll then turn our attention to the more general nature of the ring of intergers of a number field, the discriminant, regulator, class number, etc... (all of the goodies!). We will prove the fundamental theorem on fractional ideals, Dirichlet's "unit theorem," and the Dedekind-Weber Theorem, then turn to a more "modern" approach via places and adeles/ideles. If time allows, we'll then consider the zeta function of a number field and also connections to global function fields.
Students are expected to hone their mathematical skills for constructing and communicating proofs, as well as gain a deeper understanding of number theory and the role algebra plays in the investigations of the integers.
Grading Scale
Grades for section 1 will be based on homework and the final exam. The weights for these are 70% and 30%, respectively.
Final Exam
The final exam will be held Wednesday, April 28 from 10:00-11:50. It will be an in-person examination taking place in DuSable 306. Here is a list of the examination questions.
Homework
Homework will be assigned somewhat irregularly; I will usually have one assignment per week. I'll announce assignments (and due dates) in class and also post them to this webpage. You are free to work with other students on the homework; in fact, this is encouraged. Sloppy and/or illegible work will be returned back with no credit! Your homework is something of which you should be proud (notice how I didn't end with a preposition there). Expect to spend lots of time on it.
Homework Assignments
- Week #1
- Week #2
- Week #3
- Week #4
- Week #5
- Week #6
- Week #7
- Week #8
- For "Week #9" do a complete write up of answers to the questions posed on the March 24 (and 26) slides below. (Due Friday, April 2)
- Week #9 (pushed back to really be "Week #10" homework, due Friday, April 9)
- "Week #10"
Slides
- Monday, January 11
- Wednesday, January 13
- Friday, January 15
- Wednesday, January 20
- Friday, January 22
- Monday, January 25
- Wednesday, January 27
- Friday, January 29
- Monday, February 1
- Wednesday, February 3
- Friday, February 5
- Monday, February 8
- Wednesday, February 10
- Friday, February 12
- Monday, February 15
- Wednesday, February 17
- Friday, February 19 (discussion of Homework 5, #4)
- Monday, February 22
- Wednesday, February 24 (We didn't get through all these; we continued on Friday, February 26.)
- Friday, February 26
- Monday, March 1
- Wednesday and Friday, March 3 and 5
- Monday, March 8 (carried over some to March 10)
- Wednesday, March 10
- Monday, March 15
- Wednesday, March 17 (to resume Monday, March 29)
- Wednesday, March 24 (carried over some to March 26)
- Monday, March 29
- Friday, April 2
- Wednesday, April 7
- Friday, April 9
- Monday, April 12
- Wednesday, April 14
- Friday, April 16
Handouts
- I've TeXed up a compilation of some useful results from algebra (mainly from MATH 620).
- Here is a writeup of "basic" facts about algebraic integers
- This explicitly worked out example will help with the second homework assignment.
- Here is a proof of our "Fundamental Theorem."
- This handout contains goodies such as the definition of order at a prime ideal and the Chinese Remainder Theorem. Good stuff.
- I've written up a proof of Dedekind's theorem for your reading enjoyment. Homomorphisms, kernels, commutative diagrams -- the budding algebraist in you squeals with delight.
- I have typed up notes on an application of Minkowski's Theorem which describes how one can show that each ideal class has a representative ideal with fairly small norm and introduces the Minkowski constant. This is also the handout referred to in homework #5.
- For those who didn't take MATH 680 last semester, here is a writeup on some basic facts about infinite products which comes in handy when one discusses the Euler product formula.
- For those who haven't seen this sort of thing before this handout describes in great, gory detail how one can use Cauchy sequences to get a topological completion of the field of rational numbers. The exact same procedure may be used on any field with an absolute value.
- Here is a handout on the adele ring, in a format more convenient than the slides.
- In the same vein, here is a writeup of some basic stuff about absolute values on the field of rational numbers, and this covers material on places of an arbitrary number field or function field.
- This handout discusses the product formula for both number fields and function fields.
- By popular demand, I've written a short handout discussing a few examples of places over function fields.
DRC Statement
If you need an accommodation for this class, please contact the Disability Resource Center as soon as possible. The DRC coordinates accommodations for students with disabilities. It is located on the 4th floor of the Health Services Building, and can be reached at 815-753-1303 or drc@niu.edu. Also, please contact me privately as soon as possible so we can discuss your accommodations. Please note that you will not be required to disclose your disability, only your accommodations. The sooner you let me know your needs, the sooner I can assist you in achieving your learning goals in this course.
Academic Conduct
Academic honesty and mutual respect (student with student and instructor with student) are expected in this course. Mutual respect means being on time for class and not leaving early, being prepared to give full attention to class work, not reading newspapers or other material in class, not using cell phones or pagers during class time, and not looking at another student's work during exams. Academic misconduct, as defined by the Student Judicial Code, will not be treated lightly.
Coronavirus
Your health and safety are my No. 1 priority. We are all members of the Huskie community, and we owe it to each other to protect ourselves and each other. When I come to class, I’ll be wearing a face covering. I expect you to do the same. I also expect you to monitor your health, and you should stay home if you’ve been exposed to someone who recently tested positive for COVID-19 or if you develop any symptoms that might be related to COVID-19. If you do have symptoms, stay home and contact NIU’s COVID helpline (815-753-0444) to report your symptoms and get advice.
Homer does math!
Yes indeed, there's plenty of math humor to be found in the Simpsons. Just look and see!
Last update: April 22, 2021