NMAG456
Summer semester 2024/25
Tuesday 12:20 lecture
Course description:
This course will offer an introduction to the study of quadratic forms over fields. We shall cover topics such as multiplicative forms, Hilbert's 17th problem, the level of fields (rings), the Pythagoras number and the u-invariant. Depending on the time and the preferences of participants, other topics might be added (or expanded upon). This course does not require Quadratic forms and class fields I (NMAG455) as a prerequisite.
Notes:
Notes from 2023/24 (in English, by Nicolas Daans)
Outline:
Lecture 1 (18/02): Introduction - Orthogonal sums [See Notes - Lecture 1 & Pfister - Ch. 1, §1]
Lecture 2 (25/02): The representation theorem [See Notes - Pfister - Ch. 1, §2]
Lecture 3 (04/03): The subform theorem - Witt's theorems [See Notes - Ch. 2 & Pfister - Ch. 1, §2-3, Ch. 2, §1]
Lecture 4 (11/03): Multiplicative forms - Hurwitz-Radon theorem [See Pfister - Ch. 2, §2 and Lam - Ch. V, 4-5]
Lecture 5 (18/03): Classification of multiplicative forms [See Pfister - Ch. 2, §3]
Lecture 6 (25/03): The level of the fields and rings [See Pfister - Ch. 3 §1-2]
Lecture 7 (01/04): Ordered and real fields [See Pfister - Ch. 6, §1]
Lecture 8 (08/04): Hilbert's problem [See Pfister - Ch. 6, §2-3]
Lecture 9 (15/04): Quantitative bounds for the number of squares [See Pfister - Ch. 6, §3]
Lecture 10 (22/04) The Pythagoras number of fields [See Pfister - Ch. 7]
Lecture 11 (29/04) The Pythagoras number of rings - u-invariant [See Pfister - Ch. 7, Ch. 8]
Lecture 12 (06/05) The general u-invariant [See Pfister - Ch. 8]
Lecture 13 (20/05) u-invariant - Systems of quadratic forms - Leep's theorem [See Pfister - Ch. 8-9]
Recommended literature:
Quadratic Forms with Applications to Algebraic Geometry and Topology - A. Pfister,
Introduction to Quadratic Forms over Fields - T. Y. Lam