Skip to main content

MA4J4 content

Content:
Quadratic and symmetric bilinear forms over fields
The Witt group W(F) of a field F, chain lemma, cancellation and presentation of W(F)
Classification of quadratic forms over Q, R, finite fields and algebraically closed fields
Stable classification of symmetric bilinear forms over the integers
Formally real fields, signatures, sums of squares, torsion in W(F), transfer
Extension to Dedekind domains, Milnor's exact sequence

Aims:
Quadratic forms are homogeneous polynomials of degree 2 in several variables. They appear in many parts of mathematics where one reduces the classification of certain objects to the classification of quadratic forms. This happens for instance in algebra (quaternion algebras), in manifold theory (cohomology intersection form), in Lie theory (Killing form), in lattice theory (e.g., sphere packing problems), in number theory (sums of squares formulas, quadratic reciprocity) etc. The aim of this module is to understand the classification of quadratic forms over fields (e.g., field of rational numbers, finite fields) and certain rings (e.g., the integers) and to understand the relationship between properties of quadratic forms and properties of the fields in question.

Objectives: By the end of the module the student should be able to:
Understand the use of Witt groups in the classification of quadratic forms
Compute Witt groups in easy examples
Decide whether two given quadratic forms (over Q, R, F_q etc) are equivalent
Relate properties of fields to properties of quadratic forms and vice versa

Books:
Kazimierz Szymiczek, Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms. 1997. xii+486 pp. ISBN: 90-5699-076-4
(Elementary text with lots of exercises, covers part of the module)

John Milnor, Dale Husemoller, Symmetric Bilinear Forms. 1973. viii+147 pp
(Great text, covers everything in the module but no exercises)

T Y Lam, Introduction to Quadratic Forms over Fields. 2005. xxii+550 pp. ISBN: 0-8218-1095-2
(Covers everything in the module and much more with lots of exercises)