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Produktbild: Quadratic and Hermitian Forms
Band 270

Quadratic and Hermitian Forms

Aus der Reihe Grundlehren der mathematischen Wissenschaften

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Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

10.12.2011

Verlag

Springer Berlin

Seitenzahl

422

Maße (L/B/H)

23,5/15,5/2,4 cm

Gewicht

656 g

Auflage

Softcover reprint of the original 1st ed. 1985

Sprache

Englisch

ISBN

978-3-642-69973-3

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

10.12.2011

Verlag

Springer Berlin

Seitenzahl

422

Maße (L/B/H)

23,5/15,5/2,4 cm

Gewicht

656 g

Auflage

Softcover reprint of the original 1st ed. 1985

Sprache

Englisch

ISBN

978-3-642-69973-3

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

Email: GPSR Kontakt

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  • Produktbild: 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 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.