Memorial University of Newfoundland

Atlantic Association for Research in the Mathematical Sciences

Atlantic Algebra Centre

Mini-course

Quadratic Forms over Fields

by

Professor Erhard Neher

University of Ottawa

November 6 - 10, 2023

 

Abstract: The mini-course will give an introduction to quadratic forms over fields of characteristic not 2. The choice of topics is mostly standard and is covered in detail in the references below. Because of time constraints, very few results will be proven. The prerequisite is an advanced undergraduate algebra and linear algebra course. A detailed course description follows. 

Lecture I "Basic concepts": Quadratic forms versus symmetric bilinear forms, isometries, orthogonal group, reflections, , regular (= nondegenerate) forms, Cartan-Dieudonné Theorem, orthogonality,  hyperbolic spaces, diagonalization, Witt cancellation and Witt decomposition.

Lecture II "Quadratic forms under algebraic extensions": Base field extension of quadratic forms, Springer's Odd Degree Extension Theorem, tensor products of quadratic forms, Scharlau transfer, Frobenius reciprocity, Norm principles by Scharlau and by Knebusch. Time permitting, I will discuss extensions of these results to quadratic forms over rings.

Lecture III "Witt rings”: Construction of the Witt ring, invariants, , fundamental ideal,  examples of Witt rings, restriction, some structure theorems of Witt rings (Pfister, Milnor, Merkurjev), outlook.

References:

1. R. Elman, N. Karpenko, and A. Merkurjev, The algebraic and geometric theory of quadratic forms, Amer. Math. Soc. Colloq. Publ. 56,  2008.
2. T. Y. Lam, An Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, Amer. Math. Soc., 2005.
3. W. Scharlau, Quadratic and Hermitian Forms, Grundlehren der Mathematischen Wissenschaften 270, Springer-Verlag, 1985.

 

Monday November 6, 1 - 2:15 pm, room ED-4011

Wednesday November 8, 1 - 2:15 pm, room ED-4011

Friday November 10, 1 - 2:15 pm, room ED-4011