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

Cover......Page 1
Half Title......Page 4
Title Page......Page 6
Copyright Page......Page 7
Table of Contend......Page 8
Preface......Page 14
PART I: Bilinear spaces......Page 16
1.1 Affine and orthogonal geometry......Page 18
1.2 Overview......Page 22
1.3 Prerequisites......Page 25
1.4 Exercises......Page 27
2.1 Basic concepts and examples......Page 30
2.2 Characterization of reflexive bilinear spaces......Page 35
2.3 Exercises......Page 36
3.1 Bilinear spaces and congruence of matrices......Page 40
3.2 Determinant of a bilinear space......Page 44
3.3 Hyperbolic plane......Page 45
3.4 Exercises......Page 47
4.1 Isometries and congruence of ma trices......Page 50
4.2 Three invariants of isometry......Page 56
4.3 Exercises......Page 59
5.1 Radicals, matrices and dual spaces......Page 62
5.2 The orthogonal complement theorem......Page 66
5.3 Classification of singular spaces......Page 72
5.4 Exercises......Page 74
6: Diagonalization of bilinear spaces......Page 78
6.1 Symmetric spaces over fields of characteristic different from two......Page 79
6.2 Alternating spaces over arbitrary fields......Page 80
6.3 Symmetric spaces over fields of characteristic two......Page 83
6.4 Representation criterion......Page 85
6.5 Exercises......Page 87
7.1 Symmetries of a bilinear space......Page 90
7.2 Witt’s prolongation theorem......Page 92
7.3 Application of Witt’s cancellation: Inertia theorem......Page 97
7.4 Exercises......Page 100
8.1 Dyadic changes in orthogonal bases......Page 102
8.2 Dyadic changes in diagonalizations......Page 106
8.3 Exercises......Page 113
9.1 Real and nonreal fields with small square class groups......Page 116
9.2 Classification of symmetric spaces......Page 119
9.3 Exercises......Page 124
10.1 Matrix representation of the isometry group......Page 128
10.2 Orthogonal group......Page 133
10.3 Symplectic group......Page 140
10.4 Exercises......Page 141
PART II: Witt rings......Page 144
11.1 Isotropic planes......Page 146
11.2 Direct orthogonal sums......Page 150
11.3 Metabolic and hyperbolic spaces......Page 155
11.4 Exercises......Page 159
12.1 Existence of Witt decomposition......Page 162
12.2 Index of isotropy......Page 165
12.3 Uniqueness of Witt decomposition......Page 170
12.4 Exercises......Page 174
13.1 Similarity of symmetric spaces......Page 178
13.2 Witt group of a field......Page 182
13.3 Exercises......Page 187
14.1 Tensor product of vector spaces......Page 190
14.2 Tensor product of bilinear spaces......Page 198
14.3 Exercises......Page 204
15.1 Witt ring and the fundamental ideal......Page 206
15.2 Discriminant and the square of fundamental ideal......Page 211
15.3 Exercises......Page 218
16.1 Quadratic forms......Page 222
16.2 Quadratic forms and bilinear spaces......Page 228
16.3 Witt ring of quadratic forms......Page 237
16.4 Exercises......Page 239
17.1 Multiplicative properties......Page 244
17.2 The level of a nonreal field......Page 252
17.3 Witt ring of a nonreal field......Page 256
17.4 Exercises......Page 260
18.1 Formally real fields......Page 264
18.2 Ordered fields......Page 270
18.3 Total signature......Page 280
18.4 Exercises......Page 285
19.1 Prime ideals of W(K) and orderings of the field K......Page 290
19.2 Pfister’s local-global principle......Page 297
19.3 Units and zero divisors in Witt rings......Page 306
19.4 Pythagorean fields......Page 310
19.5 Exercises......Page 313
20.1 Equivalence of fields with respect to quadratic forms......Page 318
20.2 Witt equivalence of fields......Page 328
20.3 Exercises......Page 332
PART III: Invariants......Page 336
21.1 Elementary basic concepts......Page 338
21.2 Central simple algebras......Page 344
21.3 Hamilton quaternions......Page 347
21.4 Exercises......Page 351
22.1 Construction......Page 356
22.2 Isomorphisms of quaternion algebras......Page 363
22.3 Bilinear space of quaternions......Page 369
22.4 Exercises......Page 375
23.1 Tensor product of algebras......Page 380
23.2 Internal direct product of subalgebras......Page 389
23.3 The Hasse algebra......Page 395
23.4 Exercises......Page 400
24.1 The reciprocal a lgebra......Page 402
24.2 Brauer group of a field......Page 404
24.3 Wedderburn’s uniqueness theorem......Page 408
24.4 Exercises......Page 414
25.1 Hasse invariant......Page 418
25.2 Witt invariant......Page 422
25.3 Arason-Pfister property......Page 428
25.4 Harrison’s criterion......Page 429
25.5 Exercises......Page 435
A.l Symbols......Page 440
A.2 Symbolic Hasse and Witt invariants......Page 443
A.3 Exercises......Page 450
B.l Formai power series......Page 454
B.2 Pfister forms......Page 458
B.3 Annihilators of Pfister forms......Page 462
B.4 Ground field extensions......Page 465
B.5 Stable Witt rings......Page 469
B.6 Reduced Witt rings......Page 471
B.7 u-invariant......Page 474
B.8 Pfister ideals......Page 477
B.9 Fields with few quaternion algebras......Page 478
B.I0 Witt equivalence of fields......Page 481
Bibliography......Page 486
Notation......Page 494
Index......Page 497