529 69 10MB
English Pages 167 Year 1970
Table of contents :
Preface
Table of Contents
Chapter 1. The Theory of Witt
1.1. Bilinear forms and quadratic forms
1.2 Quadratic forms for charateristics not 2
1.3 Direct sum and orthogonal decomposition
1.4 Hyperbolic planes
1.5 Witt's Theorem. Witt decomposition.
1.6. The Grothendieck ring and the Witt ring
Appendix to Chapter 1.
Chapter 2. The Theory of Pfister
2.1 Multiplicative forms and applications to the structure of the Witt ring.
2.2 The method of transfer
2.3 Pfister's LocalGlobal Principle
2.4 Representation of definite functions as sums of squares
2.5 The theorems of Cassels. Fields of prescribed level.
Appendix to Chapter 2
Chapter 3. Simple Algebras and Clifford Algebras
3.1 Wedderburn Theory
3.2 The Brauer Group
3.3 Clifford alrebras
3.4 Quaternion algebras
3.5. The graded Brauer group. The Witt invariant and the Hasse invariant.
3.6. The Spin group and the Spinor norm
Appendix to Chapter 3.
Chapter 4. Classification Theory
4.1 Resumé and general results
4.2 Quadratic forms over local fields
4.3 Fields with only one nontrivial quaternion algebra
4.4 Results from global class field theory
4.5 Quadratic forms over global fields. The HasseMinkowski theorem.
Appendix to Chapter 4
Appendix. Quadratic Forms and Galois Cohomology
Bibliography
COPYRIGHT 1970 This book, or any parts thareof, may not be reproduced in any form without written permission from the author.

PREFACI:
These notes grew out of a course on quadratic forms
given at Queen's University during the academic year 1968/69.
While the principal books on quadratic forms
(O'Meara, Eichler, Jones, see also BorevitchShafarevitch) treat the theory of quadratic forms more or less as a (particularly interesting) part of algebraic number theory, it was my aim to present it as a theory in its own right.
This point of view seemed the more advisable
since during the past few years the general theory of quadratic forms over fields has made rapid progress, mainly through the work of A. Pfister.
1· am very grate
ful to A. Pfister and£. Witt for allowing ~e to include unpublished results of theirs in these notes.
For useful
discussions and correspondence I wish to thank,
W.D. Geyer, G. Harder, M. Knebusch, M. Kneser, G.A. Maxwell and J. Tits.
Finally, my thanks go to
P. Ribenboim for urging me to write down these notes, and to G. Edwards for help with the preparation of the manuscript. Winfried Scharlau
Queen's University Kingston, Ontario 12th August, 1969

].

TABLE OF CONTENTS Page
i
PREFACE TABLE OF CONTENTS CHAPTER I:
ll
THE THEORY OF WITT
1.1
Bilinear forms and quadratic forms
1
1.2
Quadratic forms for characteristic not 2
6
1.3
Direct sum and orthogonal decomposition
11
1,4
Hyperbolic planes
17
1.5
Witt's theorem; Witt decomposition
21
1.6
The Grothendieck Ring and the Witt Ring
27
Appendix to Chapter CHAPTER II:
THE THEORY
or
I
37
PFISTER
Multiplicative forms and applications to the structure of the Witt ring
38
2.2
The method of transfer
45
2.3
Pfister's localglobal principle
53
2.4
Representation of definite functions as sums of squares
57
The theorems of Cassels. prescribed level.
62
2.1
2.5
Fields of
Appendix to Chapter I I CHAPTER III:
7 (1
SIMPLE ALGEBRAS AND CLIFFORD ALGEBRAS
3,1
Wedderburn theory
75
3.2
The Brauer group
80
3.3
The Clifford algebra
8 t)r
3.4
Quaternion algebras
96
 ii 
CHAPTER III:
Cont'd
3.5
The graded Brauer group. The Witt invariant and the Hasse invariant.
1n
3.6
The Spin group and the spinor norm
1 () 8
Appendix to Chapter III
116
CHAPTER IV:
CLASSIFICATION THEORY
4.1
Resum& and general results
117
q.2
Quadratic forms over local fields
121
4.3
Fields with only one nantrivial quaternion alRebra
127
4.4
Results from global class field theory
132
4. 5
Quadra. tic forms 0\ er global fields HasseMinkowski theorem.
136
1
fj
'Tl1e
Appendix to Chapter IV APPENDIX:
QUADRATIC FORMS AND GALOIS COHOMOLOGY
149 1 ;){ '),
BIBLIOGRAPHY

l4E
"
 iii 
CHAPTEF 1
THE THEORY OF WITT
1.1. Bilinear forms and auadratic forms
'·Let
K be a field and
V
a finitedimensional
Kvectorspace.
Definition 1.1.1. b:
V X V~ K
(i)
A bilinear form on
b(x + x', y + y') = b(x, y) + b(x', y) + I
y',
I
I
b(ax, By) = aSb(x, y)
x, x , y,
for all
x, ye V
a, 8eK .
is called symmetri~__ i~
x, y
for all
E V •
and al 1 b
is a mapping
such that
b(x, y ) + b(x , y )
(ii)
V
b(x, y)
= b(y, x)
for all
E V.
For example, the standard inner product on
!Rn
is a
symmetric bilinear form. The pair
(V, b)
consisting of the vectorspace
the symmetric bilinear form (symmetric) bilinear space

b
will be called a
(over  1 ·.,.,
K) .
V and
Thus a bilinear space is a generalization of "d space and much of Euclidean geometry ordinary E:ucl,1. ean can
be done ¾n general bilinear spaces.
Definition 1.i.2. ( i)
(V, b)
Let
x, y E V
·we call two vectors b(x, y)
= b(y,
M, N
V
C
XE
M,
:
X
.l.
(ii) Let .l..
v1
vl
N.
y,
be a linear subspace of
temma 1.1. 3
I
{x} ..L Vl}
V.
Then,
is called the orthogonal
v1 . V.
is a linear subspace of
x E
vt,
i.e.
a EK, b(ax,
i.e.
Let for all
b(x + y, z)
Definition 1.1.4.
Let
x, y z
y)
b(x,
Theh one has for
b(y, z) = 0
for all
etc .
M ..L N'
= {x E V
Let
b(x, y) = 0
Orthogonality will be denoted by
complement of
Proof.
orthogonal if
We call two subsets
orthogonal if
Ye .l.
= 0.
x)
be a bilinear space.
y) ..L
E Vl
e v1 . O,
i.e.
(V, b)
 2 
y
= O for all
= ab(x, •
, i.e.
y)
=
E
0,
b(x, z) = O,
Then one has for all J..
X
+ y E Vl
be a bilinear space.
v1 .
VJ_
is called the radical of
( V, b) .
nondegenerate if the radical is Let
V* = HomK(V, K)
The bilinear form
v~
V;':
for all
~
x, y EV.
The bilinear s_race
nondegenerate if and only if
(V, b)
V,
defines a homomorphism
b
Proposition l'.1.5.
Proof.
( V, b)
0,
denote the dual space of
yb(x)(y) = b(x,y)
i.e.
\'le call
yb
(V, b)
is
is an isomorphism.
is nonde~enerate
'\fxEV,x.f0,3yEV
such that
Yb is injective ~
is an isomorphism since
Yb
b(x,y)*=O
dim V = dim V* .
Definition 1.1.6.
Let
b
symmetric) bilinear form. qb(x) = b(x, x) A map
q
be a (not necessarily The map
is called the quadratic form of
V~ K
J_$
exists a bilinear form
b.
called a quadratic form if there b
s.t.
consisting of a vectorspa