NMAG456
Summer semester 2020/21
Monday 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.
Outline:
Lecture 1 (08/03): Slides + Video (Introduction - Orthogonal sums - Isotropic forms) [See Pfister - Ch. 1, §1]
Lecture 2 (15/03): Slides + Video (Cassel's Theorems - Substitution Principle) [See Pfister - Ch. 1, §2-3]
Lecture 3 (22/03): Whiteboard + Video (Witt's Theorems - Witt Ring) [See Pfister - Ch. 2, §1]
Lecture 4 (29/03): Slides + Video (Guest speaker - Kristýna Zemková)
Lecture 5 (12/04): Whiteboard + Video (Multiplicative Forms - Hurwitz-Radon Theorem) [See Pfister - Ch. 2, §2 and Lam - Ch. V, 4-5]
Lecture 6 (19/04): Whiteboard + Video (Classification of Multiplicative Forms) [See Pfister - Ch. 2, §3]
Lecture 7 (26/04): Whiteboard + Video (Ordered and Real Fields) [See Pfister - Ch. 6, §1]
Lecture 8 (03/05): Whiteboard + Video (Hilbert's Problem) [See Pfister - Ch. 6, §2-3]
Lecture 9 (10/05): Whiteboard + Video (The level of the fields and rings - The Pythagoras Number) [See Pfister - Ch. 3 §1-2, Ch. 7 §1]
Lecture 10 (17/05) Whiteboard + Video (The Pythagoras Number of the field and ring) [See Pfister - Ch. 7]
Lecture 11 (24/05) Whiteboard + Video (The u-invariant) [See Pfister - Ch. 8]
Recommended literature:
Quadratic Forms with Applications to Algebraic Geometry and Topology - A. Pfister,
Introduction to Quadratic Forms over Fields - T. Y. Lam
Additional literature:
Quadratic and Hermitian Forms - W. Scharlau ( A general reference, more advanced material)
The Algebraic and Geometric Theory of Quadratic Forms - N. Karpenko, R. Elman, A. Merkurjev (A general reference, more advanced material)
Hurwitz-Radon Theorem - in "Problems and Theorems in Linear Algebra" by V. V. Prasolov (Lecture 5)
Around Hilbert's 17th Problem - K. Schmüdgen (Lecture 6)
An introduction to real algebra - T. Y. Lam (Lectures 7, 9)
An (almost trivial) local-global principle for the representation of –1 as a sum of squares in an arbitrary commutative ring - L. Bröcker, A. Dress, and R. Scharlau (The Level of Rings)
Pythagoras Numbers of Fields - D. W. Hoffmann (The Pythagoras Number of Fields)
The Pythagoras number of some affine algebras and local algebras - T.Y. Lam, M.D. Choi, Z.D. Dai (The Pythagoras Number of Rings)
On the Pythagoras number of orders in totally real number fields - R. Scharlau (The Pythagoras Number of Rings)