Bilinear Algebra: An Introduction to the Algebraic Theory of Quadratic Forms (Algebra, Logic and Applications) [1 ed.] 9056990764, 9789056990763

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Table of Contend
Preface
PART I: Bilinear spaces
1: Introduction
1.1 Affine and orthogonal geometry
1.2 Overview
1.3 Prerequisites
1.4 Exercises
2: Bilinear spaces
2.1 Basic concepts and examples
2.2 Characterization of reflexive bilinear spaces
2.3 Exercises
3: Bases and matrices of bilinear spaces
3.1 Bilinear spaces and congruence of matrices
3.2 Determinant of a bilinear space
3.3 Hyperbolic plane
3.4 Exercises
4: Isometries of bilinear spaces
4.1 Isometries and congruence of ma trices
4.2 Three invariants of isometry
4.3 Exercises
5: Nonsingular bilinear spaces
5.1 Radicals, matrices and dual spaces
5.2 The orthogonal complement theorem
5.3 Classification of singular spaces
5.4 Exercises
6: Diagonalization of bilinear spaces
6.1 Symmetric spaces over fields of characteristic different from two
6.2 Alternating spaces over arbitrary fields
6.3 Symmetric spaces over fields of characteristic two
6.4 Representation criterion
6.5 Exercises
7: W itts cancellation theorem
7.1 Symmetries of a bilinear space
7.2 Witts prolongation theorem
7.3 Application of Witts cancellation: Inertia theorem
7.4 Exercises
8: Witts chain isometry theorem
8.1 Dyadic changes in orthogonal bases
8.2 Dyadic changes in diagonalizations
8.3 Exercises
9: Symmetric spaces over some fields
9.1 Real and nonreal fields with small square class groups
9.2 Classification of symmetric spaces
9.3 Exercises
10: Isometry groups
10.1 Matrix representation of the isometry group
10.2 Orthogonal group
10.3 Symplectic group
10.4 Exercises
PART II: Witt rings
11: Metabolic and hyperbolic spaces
11.1 Isotropic planes
11.2 Direct orthogonal sums
11.3 Metabolic and hyperbolic spaces
11.4 Exercises
12: Witt decomposition of symmetric spaces
12.1 Existence of Witt decomposition
12.2 Index of isotropy
12.3 Uniqueness of Witt decomposition
12.4 Exercises
13: Witt group
13.1 Similarity of symmetric spaces
13.2 Witt group of a field
13.3 Exercises
14: Tensor products
14.1 Tensor product of vector spaces
14.2 Tensor product of bilinear spaces
14.3 Exercises
15: Witt ring
15.1 Witt ring and the fundamental ideal
15.2 Discriminant and the square of fundamental ideal
15.3 Exercises
16: Quadratic forms
16.1 Quadratic forms
16.2 Quadratic forms and bilinear spaces
16.3 Witt ring of quadratic forms
16.4 Exercises
17: Pfister forms
17.1 Multiplicative properties
17.2 The level of a nonreal field
17.3 Witt ring of a nonreal field
17.4 Exercises
18: Formally real fields and ordered fields
18.1 Formally real fields
18.2 Ordered fields
18.3 Total signature
18.4 Exercises
19: Prime ideals of the Witt ring
19.1 Prime ideals of W(K) and orderings of the field K
19.2 Pfisters local-global principle
19.3 Units and zero divisors in Witt rings
19.4 Pythagorean fields
19.5 Exercises
20: Witt equivalence of fields
20.1 Equivalence of fields with respect to quadratic forms
20.2 Witt equivalence of fields
20.3 Exercises
PART III: Invariants
21: Algebras
21.1 Elementary basic concepts
21.2 Central simple algebras
21.3 Hamilton quaternions
21.4 Exercises
22: Quaternion algebras
22.1 Construction
22.2 Isomorphisms of quaternion algebras
22.3 Bilinear space of quaternions
22.4 Exercises
23: Tensor product of algebras
23.1 Tensor product of algebras
23.2 Internal direct product of subalgebras
23.3 The Hasse algebra
23.4 Exercises
24: Brauer group
24.1 The reciprocal a lgebra
24.2 Brauer group of a field
24.3 Wedderburns uniqueness theorem
24.4 Exercises
25: Hasse and Witt invariants
25.1 Hasse invariant
25.2 Witt invariant
25.3 Arason-Pfister property
25.4 Harrisons criterion
25.5 Exercises
Appendices
A Symbolic Hasse and Witt invariants
A.l Symbols
A.2 Symbolic Hasse and Witt invariants
A.3 Exercises
B Selected problems
B.l Formai power series
B.2 Pfister forms
B.3 Annihilators of Pfister forms
B.4 Ground field extensions
B.5 Stable Witt rings
B.6 Reduced Witt rings
B.7 u-invariant
B.8 Pfister ideals
B.9 Fields with few quaternion algebras
B.I0 Witt equivalence of fields
Bibliography
Notation
Index

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