Algebraic theory of quadratic forms and central simple algebras

Sommersemester 2024

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 discuss 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 sketch Merkurjev's proof of the Milnor conjecture in degree two, which states that every central simple algebra of exponent 2 is Brauer equivalent to the tensor product of quaternion algebras.

Time and place:

Lectures: Tuesday 12-14 in B252, Friday 12-14 in A027.
Exercises: Thursday 10-12 in B041.
On Thursday 02.05. the exercise session will take place at 16:00 - 17:30 (instead of 10-12) in B 045.

Literature:

Lam, Introduction to Quadratic forms over fields.

Exercise sheets:

Sheet 1
Sheet 2