A quadratic form is a homogeneous polynomial of degree 2. The study of quadratic forms over arbitrary fields of characteristic different from two was initiated by Ernst Witt in 1936. He showed that the properties of quadratic forms over a given field are characterised by a commutative ring, the socalled Witt ring of the field. In the 1960s Albrecht Pfister discovered the importance of stongly multiplivative forms (now called Pfister forms) in this theory. After this discovery and the fomulation of structure theorems for the Witt ring, the quadratic form theory got much more scientific attention. Since then, until Vladimir Voevodsky prooved the Milnor conjecture in 2002, it remained an open question if the graded Witt ring of any field (the graduation is induced by the powers of the fundamental ideal of the Witt ring) can be descibed in terms of generators and relations.
Quadratic form theory applies to the study of sums of squares and field orderings. Furthermore, it is related to many other areas of mathematics, in particular the study of central simple algebras and linear algebraic groups.
The chapters covered in the course are the following:
- Symmetric bilinear forms (orthogonal sum, tensor product, isotropy, Witt's Theorems, decomposition, reflections, orthogonal group, multiplicative forms, level of a field)
- Quadratic forms (correspondence with symmetric bilinear forms, complications in characteristic 2, anisotropic part, discriminant, quaternion algebras)
- Field extensions (behaviour of forms, odd degree extensions, quadratic extensions, rational function field, Cassels-Pfister Theorem, function field of a quadratic form, characterisation of Pfister forms)
- Witt rings (construction, examples, spectrum and other ring theoretic properties, field orderings and signatures, Sylvester's inertia theorem, Artin-Schreier, Pfister's Local-Global Principle)
- Invariants and Milnor's conjecture