Math 274: Topics in Algebra

Spring 2004

Instructor: T. Y. Lam

Time: MWF 9 a.m.

Room: 31 Evans

Textbook: "The Algebraic Theory of Quadratic Forms" (Addison-Wesley, 1980)

Course Description: This course will be an introduction to the algebraic theory of quadratic forms, based on the the text I wrote on the subject in the Benjamin Series. The course will be pretty self-contained; a thorough grounding in Math 250AB is probably all that is required for following the course.

The algebraic theory of quadratic forms was created by E. Witt in his influential 1937 paper in the Crelle Journal. Witt's idea was to use the geometric language of symmetric inner product spaces to study quadratic forms over arbitrary fields, at least in the case of characteristic not 2. More importantly, Witt introduced the viewpoint of dealing with the totality of nonsingular quadratic forms as a whole, which was a substantial departure from earlier treatments of "one form at a time" by his predecessors in quadratic form theory. The legacy of Witt's 1937 paper is the famous "Witt ring" W(F) of quadratic forms over a field F. It is of historic interest to note that Witt's formation of W(F) preceded by about 20 years the Atiyah-Hirzebruch formation of the K-group of vector bundles over a space, and Grothendieck general formation of the Grothendieck group of an abstract category.

The course builds on Witt's theory, and combines it with the classical Artin-Schreier theory of formally real fields to yield the Local-Global Principle of quadratic forms over such fields. The theory of Pfister forms will be presented in the second half of the course: we'll learn, for instance, how to prove that the product of two sums of 1024 squares over a field is still a sum of 1024 squares, and to prove that, if -1 is a sum of 1023 squares in a field, then it is actually a sum of 512 squares in the same field!

Of course, we will not be able to reach the "high end" of this theory, such as Voevodsky's Fields Medal work on Milnor's Conjecture in Milnor's K-theory of fields, or even the proof of the Merkuryev-Suslin Theorem on norm residue symbols. However, the course will provide firm foundations for some of these modern developments.

There will be occasional homework assignments, but there will be no exams. Students are encouraged to take the course on N/NP basis if possible.