22 I. Foundations
so u is anisotmpic (and nonzero since y, w are independent). Consequently,
5(a) is orthogonal to n, which means that
O = B(&(y + am), y + are)
: B(5y, y) + 5(B(&w, y) + B(6ry, w)) + 52B(5w, in),
where the last term is zero. Since 8
Chapter III
Quarternion Algebras and norm forms
Satya Mandal
University of Kansas, Lawrence KS 66045 USA
August 15 2013
1
Construction or Denition
Denition 1.1. Let F be any eld with char(F ) = 2 and a, b F . Dene
quarternion algebra A =
a,b
F
as follows:
Chapter II
Introduction to Witt Rings
Satya Mandal
University of Kansas, Lawrence KS 66045 USA
May 11 2013
1
Denition of W (F ) and W (F )
From now on, by a quadratic form, we mean a nonsingular quadratic form
(see page 27). As always, F will denote a eld
Chapter I
Foundations of Quadratic Forms
Satya Mandal
University of Kansas, Lawrence KS 66045 USA
Fall 2013
1
Quadratic Forms and Quadratic Spaces
In this course we assume all elds F have char(F ) = 2.
Denition 1.1. Let F be a eld.
1. A quadratic form ove
Chapter IV
The Brauer Group
Satya Mandal
University of Kansas, Lawrence KS 66045 USA
August 15 2013
1
The Brauer Group
1. As always, F would denote a eld.
2. All F algebras considered are assumed to be nite dimensional.
3. Let A be an F algebra and S be a
Chapter V
The Cliord Algebras
Satya Mandal
University of Kansas, Lawrence KS 66045 USA
Septemebr 30, 2013
1
Construction of Cliord Algebras
In this section, a quadratic space (V, q ) need not be regular.
Denition 1.1. Suppose (V, q ) is a quadratic space.
Witt Groups of Exact Categories
Witt Groups of ed category
Translated or Shifted Duality
Triangulated Witt Groups
Satya Mandal, U. Kansas
Algebra Seminar, KU
September 19, 2013
Satya Mandal, U. Kansas Algebra Seminar, KU
Triangulated Witt Groups
Witt Grou
Chapter X
Milnor K theory, Milnor Conjecture
Gersten Conjecture
Satya Mandal
University of Kansas, Lawrence KS 66045 USA
November 1, 2013
1
Pster Forms
Denition 1.1. For n elements a1 , a2 , . . . , an F dene
:= n=1 1, ai .
i
a1 , . . . , an
This form has
Chapter VII QF
Gersten-Witt Complex
Satya Mandal
University of Kansas, Lawrence KS 66045 USA
October 19, 2013
In this chapter, Section 1 is from the Book of Lam. Rest is from ([TWGI,
BW]).
1
Scharlaus Transfer
Let F K be an extension of elds.
1. For a F q