Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms

501 Pages·1997·11.99 MB·English

by  Kazimierz Szymiczek

BILINEAR ALGEBRA
ALGEBRA, LOGIC AND APPLICATIONS
A series edited by
R. Göbel
Universität Gesamthochschule, Essen, Germany
A. Macintyre
The Mathematical Institute, University of Oxford, UK
Volume 1
Linear Algebra and Geometry
A. I. Kostrikin and Yu. I. Manin
Volume 2
Model Theoretic Algebra: With Particular Emphasis on Fields,
Rings, Modules
Christian U. Jensen and Helmut Lenzing
Volume 3
Foundations of Module and Ring Theory: A Handbook for
Study and Research
Robert Wisbauer
Volume 4
Linear Representations of Partially Ordered Sets and
Vector Space Categories
Daniel Simson
Volume 5
Semantics of Programming Languages and Model Theory
M. Droste and Y. Gurevich
Volume 6
Exercises in Algebra: A Collection of Exercises in Algebra,
Linear Algebra and Geometry
Edited by A. I. Kostrikin
Volume 7
Bilinear Algebra: An Introduction to the Algebraic Theory
of Quadratic Forms
Kazimierz Szymiczek
This book is part of a series. The publisher will accept continuation orders which may
be cancelled at any time and which provide for automatic billing and shipping of each
title in the series upon publication. Please write for details.
BILINEAR ALGEBRA
An Introduction to the Algebraic Theory
of Quadratic Forms
Kazimierz Szymiczek
Silesian University, Poland
CRC Press
Taylor & Francis Group
Boca Raton London New York
CRC Press is an imprint of the
Taylor & Francis Group, an informa business
A TAYLOR & FRANCIS BOOK
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British Library Cataloguing in Publication Data
Szymiczek, Kazimierz
Bilinear algebra : an introduction to the algebraic theory
of quadratic forms. - (Algebra, logic and applications ; v.
7)
1. Bilinear forms 2. Forms, Quadratic
1. Title
512.9’42
C on ten ts
Preface vil
PART I Bilinear spaces 1
1 Introduction 3
1.1 Affine and orthogonal geometry............................................. 3
1.2 Overview ................................................................................ 7
1.3 Prerequisites .......................................................................... 10
1.4 Exercises ................................................................................ 12
2 Bilinear spaces 15
2.1 Basic concepts and examples ................................................ 15
2.2 Characterization of reflexive bilinear spaces .................. 20
2.3 Exercises ................................................................................ 21
3 Bases and matrices of bilinear spaces 25
3.1 Bilinear spaces and congruence of matrices ........................ 25
3.2 Determinant of a bilinear space............................................. 29
3.3 Hyperbolic plane.................................................................... 30
3.4 Exercises ................................................................................ 32
4 Isometries of bilinear spaces 35
4.1 Isometries and congruence of matrices................................. 35
4.2 Three invariants of isometry................................................... 41
4.3 Exercises ................................................................................ 44
5 Nonsingular bilinear spaces 47
5.1 Radicals, matrices and dual spaces....................................... 47
5.2 The orthogonal complement theorem.................................... 51
5.3 Classification of singular spaces............................................. 57
vi Contents
5.4 Exercises ................................................................................ 59
6 Diagonalization of bilinear spaces 63
6.1 Symmetric spaces over fields of characteristic different from
two............................................................................................ 64
6.2 Alternating spaces over arbitrary fields................................ 65
6.3 Symmetric spaces over fields of characteristic two............... 68
6.4 Representation criterion......................................................... 70
6.5 Exercises ................................................................................ 72
7 Witt’s cancellation theorem 75
7.1 Symmetries of a bilinear space ............................................ 75
7.2 Witt’s prolongation theorem.................................................. 77
7.3 Application of Witt’s cancellation: Inertia theorem............. 82
7.4 Exercises ................................................................................ 85
8 Witt’s chain isometry theorem 87
8.1 Dyadic changes in orthogonal bases...................................... 87
8.2 Dyadic changes in diagonalizations...................................... 91
8.3 Exercises ................................................................................ 98
9 Symmetric spaces over some fields 101
9.1 Real and nonreal fields with small square class groups . .. 101
9.2 Classification of symmetric spaces......................................... 104
9.3 Exercises ................................................................................ 109
10 Isometry groups 113
10.1 Matrix representation of the isometry group........................ 113
10.2 Orthogonal group.................................................................... 118
10.3 Symplectic group.................................................................... 125
10.4 Exercises ................................................................................ 126
PART II Witt rings 129
11 Metabolic and hyperbolic spaces 131
11.1 Isotropic planes....................................................................... 131
11.2 Direct orthogonal sums ........................................................ 135
11.3 Metabolic and hyperbolic spaces .......................................... 140
11.4 Exercises ................................................................................ 144
Contents
12 Witt decomposition of symmetric spaces 147
12.1 Existence of Witt decomposition.......................................... 147
12.2 Index of isotropy..................................................................... 150
12.3 Uniqueness of Witt decomposition....................................... 155
12.4 Exercises ................................................................................. 159
13 Witt group 163
13.1 Similarity of symmetric spaces ............................................. 163
13.2 Witt group of a field.............................................................. 167
13.3 Exercises ................................................................................. 172
14 Tensor products 175
14.1 Tensor product of vector spaces.............................................. 175
14.2 Tensor product of bilinear spaces.......................................... 183
14.3 Exercises ................................................................................. 189
15 Witt ring 191
15.1 Witt ring and the fundamental ideal.................................... 191
15.2 Discriminant and the square of fundamental ideal............... 196
15.3 Exercises ................................................................................. 203
16 Quadratic forms 207
16.1 Quadratic forms ..................................................................... 207
16.2 Quadratic forms and bilinear spaces ..................................... 213
16.3 Witt ring of quadratic forms................................................... 222
16.4 Exercises ................................................................................ 224
17 Pfister forms 229
17.1 Multiplicative properties......................................................... 229
17.2 The level of a nonreal field...................................................... 237
17.3 Witt ring of a nonreal field ............................................ 241
17.4 Exercises ................................................................................ 245
18 Formally real fields and ordered fields 249
18.1 Formally real fields.................................................................. 249
18.2 Ordered fields.......................................................................... 255
18.3 Total signature........................................................................ 265
18.4 Exercises ................................................................................ 270