Introduction to quadratic forms over fields:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Providence, RI
American Math. Soc.
2005
|
| Series: | Graduate studies in mathematics
67 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Hier auch später erschienene, unveränderte Nachdrucke, Includes bibliographical references [S. 533 - 541] and index |
| Physical Description: | XXI, 550 S. graph. Darst. |
| ISBN: | 0821810952 9780821810958 |
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| 245 | 1 | 0 | |a Introduction to quadratic forms over fields |c T. Y. Lam |
| 264 | 1 | |a Providence, RI |b American Math. Soc. |c 2005 | |
| 300 | |a XXI, 550 S. |b graph. Darst. | ||
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Record in the Search Index
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| adam_text | Contents
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
§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
Exercises for Chapter I 22
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
Exercises for Chapter II 47
Chapter III. Quaternion Algebras and their Norm Forms 51
§1. Construction of Quaternion Algebras 51
vii
§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
Exercises for Chapter III 75
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
Exercises for Chapter IV 102
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 A^F 132
Exercises for Chapter V 140
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
Exercises for Chapter VI 183
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
Exercises for Chapter VII 226
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 Xp 271
§7. Prime Spectrum of W(F) 277
§8. Applications to the Structure of W(F) 281
§9. An Introduction to Preorderings 288
Exercises for Chapter VIII 292
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
Exercises for Chapter IX 313
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 / F 352
§6. Milnor s Higher A Groups 361
Exercises for Chapter X 372
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 {An) 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
Exercises for Chapter XI 421
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
Exercises for Chapter XII 476
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
§6. Some Open Questions 526
Exercises for Chapter XIII 531
Bibliography 533
Index 543
|
| adam_txt |
Contents
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
§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
Exercises for Chapter I 22
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
Exercises for Chapter II 47
Chapter III. Quaternion Algebras and their Norm Forms 51
§1. Construction of Quaternion Algebras 51
vii
§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
Exercises for Chapter III 75
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
Exercises for Chapter IV 102
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 A^F 132
Exercises for Chapter V 140
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
Exercises for Chapter VI 183
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
Exercises for Chapter VII 226
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 Xp 271
§7. Prime Spectrum of W(F) 277
§8. Applications to the Structure of W(F) 281
§9. An Introduction to Preorderings 288
Exercises for Chapter VIII 292
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
Exercises for Chapter IX 313
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 /"F 352
§6. Milnor's Higher A' Groups 361
Exercises for Chapter X 372
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 {An) 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
Exercises for Chapter XI 421
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
Exercises for Chapter XII 476
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
§6. Some Open Questions 526
Exercises for Chapter XIII 531
Bibliography 533
Index 543 |
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| spelling | Lam, Tsit-Yuen 1942- Verfasser (DE-588)123117704 aut Introduction to quadratic forms over fields T. Y. Lam Providence, RI American Math. Soc. 2005 XXI, 550 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Graduate studies in mathematics 67 Hier auch später erschienene, unveränderte Nachdrucke, Includes bibliographical references [S. 533 - 541] and index Forms, Quadratic Quadratische Form (DE-588)4128297-8 gnd rswk-swf Körper Algebra (DE-588)4308063-7 gnd rswk-swf Quadratische Form (DE-588)4128297-8 s Körper Algebra (DE-588)4308063-7 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4704-2108-3 Graduate studies in mathematics 67 (DE-604)BV009739289 67 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014189655&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Lam, Tsit-Yuen 1942- Introduction to quadratic forms over fields Graduate studies in mathematics Forms, Quadratic Quadratische Form (DE-588)4128297-8 gnd Körper Algebra (DE-588)4308063-7 gnd |
| subject_GND | (DE-588)4128297-8 (DE-588)4308063-7 |
| title | Introduction to quadratic forms over fields |
| title_auth | Introduction to quadratic forms over fields |
| title_exact_search | Introduction to quadratic forms over fields |
| title_exact_search_txtP | Introduction to quadratic forms over fields |
| title_full | Introduction to quadratic forms over fields T. Y. Lam |
| title_fullStr | Introduction to quadratic forms over fields T. Y. Lam |
| title_full_unstemmed | Introduction to quadratic forms over fields T. Y. Lam |
| title_short | Introduction to quadratic forms over fields |
| title_sort | introduction to quadratic forms over fields |
| topic | Forms, Quadratic Quadratische Form (DE-588)4128297-8 gnd Körper Algebra (DE-588)4308063-7 gnd |
| topic_facet | Forms, Quadratic Quadratische Form Körper Algebra |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=014189655&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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