Lecture notes for lecture 8 are now available on the course webpage.




Spring 2011 (Monday, January 31 to Friday, April 1)


  • Tue 09:05 - 09:55


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.


  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)


Detlev Hoffmann
Phone (0115) 8467142
Interests Quadratic forms, K-theory and lattices
Photo of Detlev Hoffmann


Photo of Kalyan Banerjee
Kalyan Banerjee
Photo of Heather BURKE
Heather BURKE
Photo of Biswarup Das
Biswarup Das
Photo of Andrew Davies
Andrew Davies
Photo of Daniel Kirk
Daniel Kirk
Photo of John McLeod
John McLeod
Photo of Robert Royals
Robert Royals
(East Anglia)
Photo of Nikesh Solanki
Nikesh Solanki
Photo of Luke Stanbra
Luke Stanbra
Photo of Scott Thomson
Scott Thomson
Photo of Javier Utreras
Javier Utreras


No bibliography has been specified for this course.


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