Algebraic theory of quadratic forms and Kaplansky's problem

Sommersemester 2025

This course is an introduction to the algebraic theory of quadratic forms and central simple algebras over fields. The following topics will be covered: Witt rings, Pfister forms, quadratic forms over extensions of fields, Clifford algebras, Brauer group of a field, Milnor K-theory. As an application of the developed theory, we will see the proofs of several results related to Kaplansky's problem on the possible values of two field invariants: the level and the u-invariant (note that in the case of u-invariant the problem is only partially solved). Depending on the remaining time, we will also discuss the Milnor conjecture in degree two (proven by Merkurjev), which states that every central simple algebra of exponent 2 is Brauer equivalent to the tensor product of quaternion algebras.

Time and place:

Lectures: Monday 12-14 in B251, Friday 12-14 in B252.
Exercises: Thursday 10-12 in B041 (starting from 08.05).

Literature:

T. Y. Lam, Introduction to Quadratic forms over fields.
R. Elman, N. Karpenko, A. Merkurjev, The algebraic theory of quadratic forms.
B. Kahn, Formes quadratiques sur un corps (in French, available online).