Memorial University of Newfoundland
Atlantic Association for Research in the Mathematical Sciences
Atlantic Algebra Centre
Quadratic Forms over Fields
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.
- R. Elman, N. Karpenko, and A. Merkurjev, The algebraic and geometric theory of quadratic forms, Amer. Math. Soc. Colloq. Publ. 56, 2008.
- T. Y. Lam, An Introduction to Quadratic Forms over Fields, Graduate Studies in Mathematics 67, Amer. Math. Soc., 2005.
- 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