Math 101 Topics in Algebra: Quadratic Forms

Semester: Fall 2020

Hyperbola
Graduate Course
Prof : Asher Auel
asher * auel AT dartmouth * edu
Time : Mon Wed Fri 11:45 am - 12:50 pm
X-hour Tue 12:30 - 1:20 pm
Loct : Remote
Office : Remote
Phone : No office phone!
Office
hours :
X-hour
Detailed course syllabus.

Description of course: This course will provide an introduction to the algebraic theory of quadratic forms over fields. Topics will include the elementary invariants (discriminant and Hasse invariant), classification over various fields (real numbers, finite fields, p-adic numbers, rational numbers, and rational function fields), local-global principals (e.g., Hasse-Minkowski and the Milnor exact sequence), isotropy and Witt decomposition, Witt groups, Pfister forms, and Milnor K-theory. Along the way, we will also cover the basic theory of complete discretely valued fields, orthogonal groups, and lattices. Special topics might include sums of squares (e.g., Lagrange’s theorem and Pfister’s bound for the Pythagoras number), the Milnor conjectures, the u-invariant, and trace forms, depending on the interests of the participants.

Expected background: Prior experience with linear algebra will be necessary and prior experience with field and Galois theory will be helpful but not necessary.

Homework 50%
Takehome midterm exam   20%
Final exam 30%
Grades: Your final grade will be based on homework, a midterm exam, and a final exam. All exams will be take-home.
Group work, honestly: Working with other people on mathematics is highly encouraged and fun. You may work with anyone (e.g., other students in the course, not in the course, tutors, ...) on your homework problems. If done right, you'll learn the material better and more efficiently working in groups. The golden rule is:
Work with anyone on solving your homework problems,
but write up your final draft by yourself.
Writing up the final draft is as important a process as figuring out the problems on scratch paper with your friends, see the guidelines below. If you work with people on a particular assignment, you must list your collaborators on the top of the first page. This makes the process fun, transparent, and honest. Mathematical writing is very idiosyncratic; if your proofs are copied, it is easy to tell. You will not learn (nor adhere to the Honor Principle) by copying solutions from others or from the internet.

Additional notes

X-hour: The X-hour will usually consist of office hours. On the occasional week, the X-hour will serve another purpose (e.g., extra or make-up lecture time) and I'll announce it in advance.

Homework: Weekly homework will be due each Friday. Each assignment will be posted on the syllabus page the week before it's due.

You might consider taking the opportunity to learn/practice LaTeX.

Your lowest homework score above 50% from the semester will be dropped.

Exams: The takehome midterm and final exam will be assigned. You will not be able to work together during the take-home midterm exam.




Homework guidelines: Generally, a homework problem in any math course will consist of two parts: the creative part and the write-up.

  • The creative part: This is when you "solve" the problem. You stare at it, poke at it, and work on it until you understand what's being asked, and then try different ideas until you find something that works. This part is fun to do with your friends; you can do it on the back of a napkin. If you're having trouble, even in understanding what the problem's asking, use the resources available to you: my office hours, teaching assistants' office hours, weekly tutoring sessions, etc. Ask for help as early as you can! This part should all be done on "scratch paper."

  • The write-up: Now that everything about the problem is clear in your mind, you go off by yourself and write up a coherent, succinct, and nicely written solution on clean sheets of paper. Consider this your final draft, just as in any other course. This part you should definitely NOT do with your friends.