Quadratic and Hermitian Forms

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 WittRing.- 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 forInvolutions.-
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.