MAGIC038: The algebraic theory of quadratic forms

Course details

Semester

Spring 2010
Monday, January 11th to Friday, March 19th

Hours

Live lecture hours
10
Recorded lecture hours
0
Total advised study hours
0

Description

Prerequisites

A solid foundation in algebra, including commutative rings, finite fields, and some group theory, as perhaps provided at many UK universities in 3rd year algebra courses on rings and modules or on commutative algebra, and on groups. Some knowledge in noncommutative ring theory might be helpful but isn't essential.

Syllabus

  1. Quadratic forms over general fields and their basic properties: Diagonalization, isometry, isotropy, hyperbolic forms
  2. Witt's theory: Witt cancellation, Witt decomposition
  3. The Witt ring of a field and its ring-theoretic properties
  4. The computation of the Witt ring for certain fields
  5. Orderings and formally real fields
  6. Pfister's local-global principle
  7. The fundamental ideal and the filtration of the Witt ring
  8. The Cassels-Pfister theorem
  9. Round and multiplicative forms, Pfister forms
  10. The Arason-Pfister Hauptsatz
  11. Quaternion algebras and their norm forms
  12. Basic theory of central simple algebras
  13. The Clifford algebra of a quadratic form
  14. The classical invariants of quadratic forms: dimension, discriminant, Clifford invariant
  15. Merkurjev's Theorem
  16. A first glimpse of the Milnor conjecture (Voevodsky's theorem)

Lecturer

  • DH

    Prof Detlev Hoffmann

    University
    University of Nottingham

Bibliography

No bibliography has been specified for this course.

Assessment

Attention needed

Assessment information will be available nearer the time.

Files

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Lectures

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