Introduction to quadratic forms over fields / T.Y. Lam.

Format:

Book

Author/Creator:

Lam, T. Y. (Tsit-Yuen), 1942-

Series:

Graduate studies in mathematics ; v. 67.

Graduate studies in mathematics, 1065-7339 ; v. 67

Language:

English

Subjects (All):

Forms, Quadratic.

Physical Description:

xxi, 550 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society, [2005]

Contents:

1 Quadratic Forms and Quadratic Spaces 1

2 Diagnoalization of Quadratic Forms 5

3 Hyperbolic Plane and Hyperbolic Spaces 9

4 Decomposition Theorem and Cancellation Theorem 12

5 Witt's Chain Equivalence Theorem 15

6 Kronecker Product of Quadratic Spaces 17

7 Generation of the Orthogonal Group by Reflections 18

Chapter II Introduction to Witt Rings 27

1 Definition of W(F) and W(F) 27

2 Group of Square Classes 30

3 Some Elementary Computations 33

4 Presentation of Witt Rings 39

5 Classification of Small Witt Rings 41

Chapter III Quaternion Algebras and their Norm Forms 51

1 Construction of Quaternion Algebras 51

2 Quaternion Algebras as Quadratic Spaces 55

3 Coverings of the Orthogonal Groups 63

4 Linkage of Quaternion Algebras 67

5 Characterizations of Quaternion Algebras 73

Chapter IV The Brauer-Wall Group 79

1 The Brauer Group 79

2 Central Simple Graded Algebras (CSGA) 83

3 Structure Theory of CSGA 90

4 The Brauer-Wall Group 98

Chapter V Clifford Algebras 103

1 Construction of Clifford Algebras 103

2 Structure Theorems 108

3 The Clifford Invariant, Witt Invariant, and Hasse Invariant 113

4 Real Periodicity and Clifford Modules 122

5 Composition of Quadratic Forms 127

6 Steinberg Symbols and Milnor's Group k[subscript 2]F 132

Chapter VI Local Fields and Global Fields 143

1 Springer's Theorem for C.D.V. Fields 143

2 Quadratic Forms over Local Fields 150

Appendix Nonreal Fields with Four Square Classes 167

3 Hasse-Minkowski Principle 169

4 Witt Ring of Q 174

5 Hilbert Reciprocity and Quadratic Reciprocity 178

Chapter VII Quadratic Forms Under Algebraic Extensions 187

1 Scharlau's Transfer 187

2 Simple Extensions and Springer's Theorem 191

3 Quadratic Extensions 196

4 Scharlau's Norm Principle 204

5 Knebusch's Norm Principle 206

6 Galois Extensions and Trace Forms 209

7 Quadratic Closures of Fields 218

Chapter VIII Formally Real Fields, Real-Closed Fields, and Pythagorean Fields 231

1 Structure of Formally Real Fields 231

2 Characterizations of Real-Closed Fields 240

Appendix A Uniqueness of Real-Closure 246

Appendix B Another Artin-Schreier Theorem 250

3 Pfister's Local-Global Principle 252

4 Pythagorean Fields 255

Appendix Fields with 8 Square Classes and 2 Orderings 265

5 Connections with Galois Theory 267

6 Harrison Topology on X[subscript F] 271

7 Prime Spectrum of W(F) 277

8 Applications to the Structure of W(F) 281

9 An Introduction to Preorderings 288

Chapter IX Quadratic Forms under Transcendental Extensions 299

1 Cassels-Pfister Theorem 299

2 Second and Third Representation Theorems 303

3 Milnor's Exact Sequence for W(F(x)) 306

4 Scharlau's Reciprocity Formula for F(x) 309

Chapter X Pfister Forms and Function Fields 315

1 Chain P-Equivalence 316

Appendix Round Forms 322

2 Multiplicative Forms 323

3 Introduction to Function Fields 328

4 Basic Theorems on Function Fields 334

5 Hauptsatz, Linkage, and Forms in I[superscript n] F 352

6 Milnor's Higher K-Groups 361

Chapter XI Field Invariants 375

1 Sums of Squares 376

2 The Level of a Field 379

3 Pfister-Witt Annihilator Theorem 384

4 The Property (A[subscript n]) 388

5 Height and Pythagoras Number 394

6 The u-Invariant of a Field 398

Appendix The General u-Invariant 409

7 The Size of W(F), and C-Fields 413

Chapter XII Special Topics in Quadratic Forms 425

1 Isomorphisms of Witt Rings 426

2 Quadratic Forms of Low Dimension 431

Appendix Forms with Isomorphic Function Fields 437

3 Some Classification Theorems 439

4 Witt Rings under Biquadratic Extensions 443

5 Nonreal Fields with Eight Square Classes 447

6 Kaplansky Radical and Hilbert Fields 450

7 Construction of Some Pre-Hilbert Fields 456

8 Axiomatic Schemes for Quadratic Forms 463

Chapter XIII Special Topics on Invariants 479

1 The u-Invariant of C((x,y)) 480

2 Fields of u-Invariant 6 484

3 Fields of Pythagoras Number 6 and 7 495

4 Levels of Commutative Rings 499

5 Pythagoras Numbers of Commutative Rings 514.

Notes:

Includes bibliographical references (pages 533-541) and index.

ISBN:

0821810952

OCLC:

56809496