内容

目次

1. Basic Concepts.- §1. Bilinear Forms and Quadratic Forms.- § 2. Matrix Notation.- § 3. Regular Spaces and Orthogonal Decomposition.- § 4. Isotropy and Hyperbolic Spaces.- §5. Witt’s Theorem.- §6. Appendix: Symmetric Bilinear Forms and Quadratic Forms over Rings.- 2. Quadratic Forms over Fields.- §1. Grothendieck and Witt Rings.- §2. Invariants.- § 3. Examples I (Finite Fields).- § 4. Examples II (Ordered Fields).- § 5. Ground Field Extension and Transfer.- §6. The Torsion of the Witt Group.- §7. Orderings, Pfister’s Local Global Principle, and Prime Ideals of the Witt Ring.- § 8. Applications of the Method of Transfer.- § 9. Description of the Witt Ring by Generators and Relations.- § 10. Multiplicative Forms.- §11. Quaternion Algebras.- §12. The Hasse Invariant and the Witt Invariant.- §13. The Hasse Algebra.- § 14. Classification Theorems.- §15. Examples III. Ci-fields.- §16. The u-invariant.- 3. Quadratic Forms over Formally Real Fields.- §1. Formally Real and Ordered Fields.- §2. Real Closed Fields.- §3. Hilbert’s 17th Problem and the Real Nullstellensatz.- §4. Extension of Signatures.- §5. The Space of Orderings of a Field.- §6. The Total Signature.- §7. A Local Global Principle for Weak Isotropy.- Appendix: Places, Valuations, and Valuation Rings.- 4. Generic Methods and Pfister Forms.- §1. Chain-p-equivalence of Pfister Forms.- § 2. Pfister’s Theorem on the Representation of Positive Functions as Sums of Squares.- §3. Casseis’ and Pfister’s Representation Theorems.- §4. Applications: Fields of Prescribed Level. Characterization of Pfister Forms.- §5. The Function Field of a Quadratic Form and the Main Theorem of Arason and Pfister.- §6. Generic Zeros and Generic Splitting.- §7. Knebusch’s Filtration of the Witt Ring.- 5. Rational Quadratic Forms.- § 1. Symmetric Bilinear Forms and Quadratic Forms on Finite Abelian Groups.- §2. Gaussian Sums for Quadratic Forms on Finite Abelian Groups.- §3. The Witt Group of1.- §4. The Witt Group of 2.- §5. Gauss’ First Proof of the Quadratic Reciprocity Law.- §6. Quadratic Forms over the p-adic Numbers.- §7. Hilbert’s Reciprocity Law and the Hasse-Minkowski Theorem.- §8. Calculation of Gaussian Sums.- 6. Symmetric Bilinear Forms over Dedekind Rings and Global Fields.- §1. Symmetric Bilinear Forms over Dedekind Rings.- § 2. Symmetric Bilinear Forms over Discrete Valuation Rings.- §3. Symmetric Bilinear Forms over Polynomial Rings and Rational Function Fields.- §4. Symmetric Bilinear Forms over p-adic Fields.- § 5. The Hilbert Reciprocity Theorem.- § 6. The Hasse-Minkowski Theorem.- § 7. Hecke’s Theorem on the Different.- §8. The Residue Theorem.- 7. Foundations of the Theory of Hermitian Forms.- §1. Basic Definitions.- § 2. Hermitian Categories.- § 3. Quadratic Forms.- §4. Transfer and Reduction.- § 5. Hermitian Abelian Categories.- §6. Hermitian Forms over Skew Fields.- §7. Hyperbolic Forms and the Unitary Group.- §8. Alternating Forms and the Symplectic Group.- §9. Witt’s Theorem.- § 10. The Krull-Schmidt Theorem.- §11. Examples and Applications.- 8. Simple Algebras and Involutions.- §1. Simple Rings and Modules.- §2. Tensor Products.- § 3. Central Simple Algebras. The Brauer Group.- §4. Simple Algebras.- §5. Central Simple Algebras under Field Extensions. Reduced Norms and Traces.- §6. Examples.- §7. Involutions on Simple Algebras. The Classification Problem.- § 8. Existence of Involutions.- §9. The Corestriction. Existence of Involutions of the Second Kind.- § 10. An Extension Theorem for Involutions.- §11. Quaternion Algebras.- §12. Cyclic Algebras.- §13. The Canonical Involution on the Group Algebra.- 9. Clifford Algebras.- §1. Graded Algebras.- §2. Clifford Algebras.- § 3. The Spinor Norm.- §4. Quadratic Forms over Fields in Characteristic 2.- 10. Hermitian Forms over Global Fields.- §1. Hermitian Forms over Commutative Fields and Quaternion Algebras.- §2. Simple Algebras and Involutions over Local and Global Fields.- §3. Skew Hermitian Forms over Quaternion Fields.- §4. Skew Hermitian Forms over Global Quaternion Fields..- §5. The Strong Approximation Theorem.- §6. Hermitian Forms for Unitary Involutions. Statement of Results.- §7. Proof of the Weak Local Global Principle.- §8. Conclusion of the Proof.