Quadratic and Hermitian Forms

For a long time - at least from Fermat to Minkowski - the theory of quadratic forms was a part of number theory. Much of the best work of the great number theorists of the eighteenth and nineteenth century was concerned with problems about quadratic forms. On the basis of their work, Minkowski, Siegel, Hasse, Eichler and many others crea­ ted the impressive "arithmetic" theory of quadratic forms, which has been the object of the well-known books by Bachmann (1898/1923), Eichler (1952), and O'Meara (1963). Parallel to this development the ideas of abstract algebra and abstract linear algebra introduced by Dedekind, Frobenius, E. Noether and Artin led to today's structural mathematics with its emphasis on classification problems and general structure theorems. On the basis of both - the number theory of quadratic forms and the ideas of modern algebra - Witt opened, in 1937, a new chapter in the theory of quadratic forms. His most fruitful idea was to consider not single "individual" quadratic forms but rather the entity of all forms over a fixed ground field and to construct from this an algebra­ ic object. This object - the Witt ring - then became the principal object of the entire theory. Thirty years later Pfister demonstrated the significance of this approach by his celebrated structure theorems.

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.