Quadratic Forms

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 Local-Global 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 non-trivial quaternion algebra
4.4 Results from global class field theory
4.5 Quadratic forms over global fields. The Hasse-Minkowski theorem.
Appendix to Chapter 4
Appendix. Quadratic Forms and Galois Cohomology
Bibliography

Citation preview

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 Borevitch-Shafarevitch) 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 Witt for allowing ~e to include unpublished results of theirs in these notes.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

-

].

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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 local-global 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 nan-trivial quaternion alRebra

127

4.4

Results from global class field theory

132

4. 5

Quadra. tic forms 0\ er global fields Hasse-Minkowski 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 finite-dimensional

K-vectorspace.

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) .

non-degenerate if the radical is Let

V* = HomK(V, K)

The bilinear form

v~

V;':

for all

~

x, y EV.

The bilinear s_race

non-degenerate 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 non-de~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