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AuthorScharlau, Winfried. author
TitleQuadratic and Hermitian Forms [electronic resource] / by Winfried Scharlau
ImprintBerlin, Heidelberg : Springer Berlin Heidelberg, 1985
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Descript X, 422 p. online resource


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 creaยญ 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 algebraยญ 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 -- ยง1. Bilinear Forms and Quadratic Forms -- ยง 2. Matrix Notation -- ยง 3. Regular Spaces and Orthogonal Decomposition -- ยง 4. Isotropy and Hyperbolic Spaces -- ยง5. Wittโ{128}{153}s 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, Pfisterโ{128}{153}s 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. Hilbertโ{128}{153}s 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. Pfisterโ{128}{153}s Theorem on the Representation of Positive Functions as Sums of Squares -- ยง3. Casseisโ{128}{153} and Pfisterโ{128}{153}s 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. Knebuschโ{128}{153}s 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{128}{153} First Proof of the Quadratic Reciprocity Law -- ยง6. Quadratic Forms over the p-adic Numbers -- ยง7. Hilbertโ{128}{153}s 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. Heckeโ{128}{153}s 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. Wittโ{128}{153}s 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

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