Mathematics 423/502, Introduction to the algebraic theory of quadratic forms, Section 201.
January-April 2012, Hennings Building 301, MWF 13:00-13:50

  • Textbook : T.Y. Lam, Introduction to quadratic forms over fields, American Mathematical Society, 2004.

  • Course description : The theory of quadratic forms originated in the 19th century, as a part of linear algebra. Over the field of complex numbers, there is only one non-degenerate quadratic forms of an n-dimensional vector space (up to a change of coordinates). This is no longer true over the field of real numbers (this is why ellipses and hyperbolas look different in the real plane), but quadratic forms over R are also quite easy to describe, via Sylvester's Law of Inertia. Over an arbitrary field K, however, the situation is quite different. In 1937 E. Witt showed that quadratic forms can be packaged into what is now now called "the Witt ring" of K. The Witt ring is an intricate structure, which can be associated to any field K. While its construction and first properties rely only on undergraduate abstract and linear algebra, the Witt ring has been found to have rich connections with many areas of mathematics, from Galois theory and number theory to algebraic geometry and topology. Research in this area is on-going; in particular, a Fields medal was awarded in 2002 for a break-through in the theory of quadratic forms (the proof of Milnor's conjecture). This course is intended to be a gentle introduction to this subject, based on the pioneering work of E. Witt (in the 1930s) and A. Pfister (in the 1960s), following an award-winning textbook by T.-Y. Lam. Specific topics will include Witt's Decomposition and Cancellation theorems, tensor products, central simple algebras, Clifford algebras, the Hasse-Witt invariant, Springer's theorem, trace forms, Pfister forms.

  • Homework : Homework will be collected in class, usually on a bi-weekly schedule. Late homework will not be accepted. Students are allowed to consult one another concerning the homework problems, but your submitted solutions must be written by you in your own words.

    Problem set 1. Due in class Monday, January 23. Exercises for Chapter I, pp. 22-25, Problems 2, 3, 5, 10, 12, 13, 14.
    Solutions to Problem Set 1.

    Problem set 2. Due in class Wednesday, February 8. Exercises for Chapter I, Problems 8, 16, 17, 28. Exercises for Chapter II, Problems 6, 10, 18, 20.
    Solutions to Problem Set 2.

    Problem Set 3. Due in class Friday, February 17.
    Solutions to Problem Set 3.

    Problem Set 4. Due in class Friday, March 9.
    Solutions to Problem Set 4.

    Problem Set 5. Due in class Friday, March 23.
    Solutions to Problem Set 5.

    Problem Set 6. Due in class Friday, April 2.
    Solutions to Problem Set 6.

  • Evaluation : Course marks will be based on the homework (50%) and the final exam (50%). Students can use a two-sided standard size (8" by 11") note sheet on the final exam, but no other notes, books or calculators.

  • Students with disabilities : Students with documented disabilities who may need special accommodations should make an appointment with me early in the term.