Title of the course: Quadratic form theory – The Theory Over (Some) Rings
Instructor: Prof. Max Dickmann
Institution: Institut de Mathématiques de Jussieu-Paris Rive Gauche, Université de Paris and Sorbonne Université
Dates: 12-24 September 2022
Prerequisites: Field theory; basic knowledge of (commutative) ring theory; linear algebra.
Level: Graduate
Abstract: In the first week I’ll develop the basics and some highlights of the classical theory of quadratic forms over (commutative) fields of characteristic not 2 with special emphasis on ordered fields. For this I’ll use some sections of Lam’s book below. In the second week I’ll concentrate on the extension of the theory over fields to (diagonal) quadratic forms with non zero-divisor coefficients over several classes of (preordered) rings. The theory of “special groups” is the main abstract tool making possible this extension. I’ll show that the theory thus obtained applies to many classes of rings met in mathematical practice.
Language: EN
Textbooks:
For the first week:
1. T.Y. Lam, Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics, Vo. 67, AMS, 2005.
For the second week:
1. M. Dickmann, F. Miraglia, Faithfully Quadratic Rings, Memoirs of the AMS 1128, 2015.
2. M. Dickmann, F.Miraglia, H. Ribeiro, Special Groups and Quadratic Forms over Rings with non Zero-Divisor Coefficients, 60 pp., to appear (2022) in Fundamenta Mathematicae.