Introduction to Quadratic Forms over Fields

Transcription

1 Introduction to Quadratic Forms over Fields T.Y. Lam Graduate Studies in Mathematics Volume 67.. American Mathematical Society 1 " " M Providence, Rhode Island

2 Preface xi ; Notes to the Reader xvii Partial List of Notations xix Chapter I. Foundations 1 1. Quadratic Forms and Quadratic Spaces 1! 2. Diagonalization of Quadratic Forms 5 ;, 3. Hyperbolic Plane and Hyperbolic Spaces 9 I 4. Decomposition Theorem and Cancellation Theorem Witt's Chain Equivalence Theorem Kronecker Product of Quadratic Spaces Generation of the Orthogonal Group by Reflections 18 Exercises for Chapter I 22 Chapter II. Introduction to Witt Rings Definition of W(F) and W(F) Group of Square Classes Some Elementary Computations Presentation of Witt Rings Classification of Small Witt Rings 41 Exercises for Chapter II 47 Chapter III. Quaternion Algebras and their Norm Forms Construction of Quaternion Algebras 51 vii

3 viii Contents 2. Quaternion Algebras as Quadratic Spaces Coverings of the Orthogonal Groups Linkage of Quaternion Algebras Characterizations of Quaternion Algebras 73 Exercises for Chapter III 75 Chapter IV. The Brauer-Wall Group The Brauer Group Central Simple Graded Algebras (CSGA) Structure Theory of CSGA The Brauer-Wall Group 98 Exercises for Chapter IV 102 Chapter V. Clifford Algebras Construction of Clifford Algebras Structure Theorems The Clifford Invariant, Witt Invariant, and Hasse Invariant Real Periodicity and Clifford Modules Composition of Quadratic Forms Steinberg Symbols and Milnor's Group &2-F 132 Exercises for Chapter V 140 Chapter VI. Local Fields and Global Fields Springer's Theorem for C.D.V. Fields Quadratic Forms over Local Fields 150 Appendix: Nonreal Fields with Four Square Classes Hasse-Minkowski Principle Witt Ring of Q Hilbert Reciprocity and Quadratic Reciprocity 178 Exercises for Chapter VI 183 Chapter VII. Quadratic Forms Under Algebraic Extensions Scharlau's Transfer Simple Extensions and Springer's Theorem Quadratic Extensions Scharlau's Norm Principle Knebusch's Norm Principle 206

4 ix 6. Galois Extensions and Trace Forms Quadratic Closures of Fields 218 Exercises for Chapter VII 226 Chapter VIII. Formally Real Fields, Real-Closed Fields, and Pythagorean Fields Structure of Formally Real Fields Characterizations of Real-Closed Fields 240 Appendix A: Uniqueness of Real-Closure 246 Appendix B: Another Artin-Schreier Theorem Pfister's Local-Global Principle Pythagorean Fields 255 Appendix: Fields with 8 Square Classes and 2 Orderings Connections with Galois Theory Harrison Topology on Xp Prime Spectrum of W(F) Applications to the Structure of W(F) An Introduction to Preorderings 288 Chapter IX. Exercises for Chapter VIII 292 Quadratic Forms under Transcendental Extensions Cassels-Pfister Theorem Second and Third Representation Theorems Milnor's Exact Sequence for W(F(x)) Scharlau's Reciprocity Formula for F(x) 309 Exercises for Chapter IX 313 Chapter X. Pfister Forms and Function Fields Chain P-Equivalence 316 Appendix: Round Forms Multiplicative Forms Introduction to Function Fields Basic Theorems on Function Fields Hauptsatz, Linkage, and Forms in I n F Milnor's Higher K-Groups 361 Exercises for Chapter X 372

5 Chapter XI. Field Invariants Sums of Squares The Level of a Field Pfister-Witt Annihilator Theorem The Property (A n ) Height and Pythagoras Number The u-invariant of a Field 398 Appendix: The General ^-Invariant The Size of W(F), and C-Fields 413 Exercises for Chapter XI 421 Chapter XII. Special Topics in Quadratic Forms Isomorphisms of Witt Rings Quadratic Forms of Low Dimension 431 Appendix: Forms with Isomorphic Function Fields Some Classification Theorems Witt Rings under Biquadratic Extensions Nonreal Fields with Eight Square Classes Kaplansky Radical and Hilbert Fields Construction of Some Pre-Hilbert Fields Axiomatic Schemes for Quadratic Forms 463 Exercises for Chapter XII 476 Chapter XIII. Special Topics on Invariants The u-invariant of C((x,y)) Fields of u-invariant Fields of Pythagoras Number 6 and Levels of Commutative Rings Pythagoras Numbers of Commutative Rings Some Open Questions 526 Exercises for Chapter XIII 531 Bibliography 533 Index 543