Author | Scharlau, Winfried. author |
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Title | Quadratic and Hermitian Forms [electronic resource] / by Winfried Scharlau |
Imprint | Berlin, Heidelberg : Springer Berlin Heidelberg, 1985 |
Connect to | http://dx.doi.org/10.1007/978-3-642-69971-9 |
Descript | X, 422 p. online resource |
1. Basic Concepts -- ยง1. Bilinear Forms and Quadratic Forms -- ยง 2. Matrix Notation -- ยง 3. Regular Spaces and Orthogonal Decomposition -- ยง 4. Isotropy and Hyperbolic Spaces -- ยง5. Wittโs Theorem -- ยง6. Appendix: Symmetric Bilinear Forms and Quadratic Forms over Rings -- 2. Quadratic Forms over Fields -- ยง1. Grothendieck and Witt Rings -- ยง2. Invariants -- ยง 3. Examples I (Finite Fields) -- ยง 4. Examples II (Ordered Fields) -- ยง 5. Ground Field Extension and Transfer -- ยง6. The Torsion of the Witt Group -- ยง7. Orderings, Pfisterโs Local Global Principle, and Prime Ideals of the Witt Ring -- ยง 8. Applications of the Method of Transfer -- ยง 9. Description of the Witt Ring by Generators and Relations -- ยง 10. Multiplicative Forms -- ยง11. Quaternion Algebras -- ยง12. The Hasse Invariant and the Witt Invariant -- ยง13. The Hasse Algebra -- ยง 14. Classification Theorems -- ยง15. Examples III. Ci-fields -- ยง16. The u-invariant -- 3. Quadratic Forms over Formally Real Fields -- ยง1. Formally Real and Ordered Fields -- ยง2. Real Closed Fields -- ยง3. Hilbertโs 17th Problem and the Real Nullstellensatz -- ยง4. Extension of Signatures -- ยง5. The Space of Orderings of a Field -- ยง6. The Total Signature -- ยง7. A Local Global Principle for Weak Isotropy -- Appendix: Places, Valuations, and Valuation Rings -- 4. Generic Methods and Pfister Forms -- ยง1. Chain-p-equivalence of Pfister Forms -- ยง 2. Pfisterโs Theorem on the Representation of Positive Functions as Sums of Squares -- ยง3. Casseisโ and Pfisterโs Representation Theorems -- ยง4. Applications: Fields of Prescribed Level. Characterization of Pfister Forms -- ยง5. The Function Field of a Quadratic Form and the Main Theorem of Arason and Pfister -- ยง6. Generic Zeros and Generic Splitting -- ยง7. Knebuschโs Filtration of the Witt Ring -- 5. Rational Quadratic Forms -- ยง 1. Symmetric Bilinear Forms and Quadratic Forms on Finite Abelian Groups -- ยง2. Gaussian Sums for Quadratic Forms on Finite Abelian Groups -- ยง3. The Witt Group of1 -- ยง4. The Witt Group of 2 -- ยง5. Gaussโ First Proof of the Quadratic Reciprocity Law -- ยง6. Quadratic Forms over the p-adic Numbers -- ยง7. Hilbertโs Reciprocity Law and the Hasse-Minkowski Theorem -- ยง8. Calculation of Gaussian Sums -- 6. Symmetric Bilinear Forms over Dedekind Rings and Global Fields -- ยง1. Symmetric Bilinear Forms over Dedekind Rings -- ยง 2. Symmetric Bilinear Forms over Discrete Valuation Rings -- ยง3. Symmetric Bilinear Forms over Polynomial Rings and Rational Function Fields -- ยง4. Symmetric Bilinear Forms over p-adic Fields -- ยง 5. The Hilbert Reciprocity Theorem -- ยง 6. The Hasse-Minkowski Theorem -- ยง 7. Heckeโs Theorem on the Different -- ยง8. The Residue Theorem -- 7. Foundations of the Theory of Hermitian Forms -- ยง1. Basic Definitions -- ยง 2. Hermitian Categories -- ยง 3. Quadratic Forms -- ยง4. Transfer and Reduction -- ยง 5. Hermitian Abelian Categories -- ยง6. Hermitian Forms over Skew Fields -- ยง7. Hyperbolic Forms and the Unitary Group -- ยง8. Alternating Forms and the Symplectic Group -- ยง9. Wittโs Theorem -- ยง 10. The Krull-Schmidt Theorem -- ยง11. Examples and Applications -- 8. Simple Algebras and Involutions -- ยง1. Simple Rings and Modules -- ยง2. Tensor Products -- ยง 3. Central Simple Algebras. The Brauer Group -- ยง4. Simple Algebras -- ยง5. Central Simple Algebras under Field Extensions. Reduced Norms and Traces -- ยง6. Examples -- ยง7. Involutions on Simple Algebras. The Classification Problem -- ยง 8. Existence of Involutions -- ยง9. The Corestriction. Existence of Involutions of the Second Kind -- ยง 10. An Extension Theorem for Involutions -- ยง11. Quaternion Algebras -- ยง12. Cyclic Algebras -- ยง13. The Canonical Involution on the Group Algebra -- 9. Clifford Algebras -- ยง1. Graded Algebras -- ยง2. Clifford Algebras -- ยง 3. The Spinor Norm -- ยง4. Quadratic Forms over Fields in Characteristic 2 -- 10. Hermitian Forms over Global Fields -- ยง1. Hermitian Forms over Commutative Fields and Quaternion Algebras -- ยง2. Simple Algebras and Involutions over Local and Global Fields -- ยง3. Skew Hermitian Forms over Quaternion Fields -- ยง4. Skew Hermitian Forms over Global Quaternion Fields. -- ยง5. The Strong Approximation Theorem -- ยง6. Hermitian Forms for Unitary Involutions. Statement of Results -- ยง7. Proof of the Weak Local Global Principle -- ยง8. Conclusion of the Proof