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.
Work with anyone on solving your homework problems,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.
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.