MAGIC038: The algebraic theory of quadratic forms

Course details

A specialist MAGIC course


Spring 2009
Monday, January 19th to Friday, March 27th; Monday, April 27th to Monday, April 27th


Live lecture hours
Recorded lecture hours
Total advised study hours


12:05 - 12:55


The slides and the hand-out version of the tenth (and final) lecture have been put on the course webpage.
Recall that the last lecture of this course will be given this MAGIC week (week 9) on Friday:
MAGIC week 9 (Mon 16 March - Fri 30 March):
Wednesday 18 March, 9:05-9:55: Extra lecture (Lecture 9)
Friday 20 March, 12:05-12:55: Regular lecture (Lecture 10 = final lecture)
MAGIC week 10 (Mon 23 March - Fri 27 March):
No lectures.



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)


  • DH

    Professor Detlev Hoffmann

    University of Nottingham


No bibliography has been specified for this course.


Attention needed

Assessment information will be available nearer the time.


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