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Quadratic and Hermitian forms

Author: Winfried Scharlau
Publisher: Berlin [etc.] : Springer, 1985.
Series: Grundlehren der mathematischen Wissenschaften (Springer), 270.
Edition/Format:   Print book : EnglishView all editions and formats

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Document Type: Book
All Authors / Contributors: Winfried Scharlau
ISBN: 3540137246 9783540137245
OCLC Number: 801956752
Description: X, 421 p.; 24 cm.
Contents: 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.
Series Title: Grundlehren der mathematischen Wissenschaften (Springer), 270.
Responsibility: Winfried Scharlau.


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