Title of the course: Introduction to quadratic forms over fields
Instructor: Dr. Elyes Boughattas
Institution: University of Bath
Dates: 14-20 July 2025
Prerequisites: Basic theory of groups, fields, rings. Linear algebra, tensor products and definition of multivariate polynomials. I could come back to some of these notions if needed.
Level: Advanced undergraduate to graduate level.
Abstract: Quadratic forms are crucial tools for algebraists, geometers and analysts. From an algebraic point of view, the study of quadratic forms is the natural continuation of linear algebra, which studies the behaviour of linear forms.
In this lecture, I will start outlining the basic theory of quadratic forms and prove Witt’s cancellation theorem. The next step is to introduce quaternion algebras, Brauer classes and Witt rings. The final goal of the lecture is to sketch the proof of Hasse-Minkowski theorem on quadratic forms over Q – which will require a brief introduction to p-adic numbers. If time permits, throughout the lecture we will tackle some classical invariants of quadratic forms, such as their dimension, their discriminant and Hasse invariant. We will then say few words about Pythagoras numbers and touch upon the main results.
Language: English