Franklin

Basic quadratic forms / Larry J. Gerstein.

Author/Creator:
Gerstein, Larry J., author.
Publication:
Providence, Rhode Island : American Mathematical Society, [2008]
Format/Description:
Book
1 online resource (274 p.)
Series:
Graduate studies in mathematics ; v. 90.
Graduate studies in mathematics, 1065-7339 ; volume 90
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Details

Subjects:
Forms, Quadratic.
Equations, Quadratic.
Number theory.
Form/Genre:
Electronic books.
Language:
English
Contents:
""Cover""; ""Title""; ""Copyright""; ""Contents""; ""Preface""; ""Chapter 1. A Brief Classical Introduction""; ""Â1.1. Quadratic Forms as Polynomials""; ""Â1.2. Representation and Equivalence; Matrix Connections; Discriminants""; ""Exercises""; ""Â1.3. A Brief Historical Sketch, and Some References to the Literature""; ""Chapter 2. Quadratic Spaces and Lattices""; ""Â2.1. Fundamental Definitions""; ""Â2.2. Orthogonal Splitting; Examples of Isometry and Non-isometry""; ""Exercises""; ""Â2.3. Representation, Splitting, and Isotropy; Invariants u(F) and s(F)""
""Â2.4. The Orthogonal Group of a Space""""Â2.5. Witt's Cancellation Theorem and Its Consequences""; ""Â2.6. Witt's Chain Equivalence Theorem""; ""Â2.7. Tensor Products of Quadratic Spaces; the Witt ring of a field""; ""Exercises""; ""Â2.8. Quadratic Spaces over Finite Fields""; ""Â2.9. Hermitian Spaces""; ""Exercises""; ""Chapter 3. Valuations, Local Fields, and p-adic Numbers""; ""Â3.1. Introduction to Valuations""; ""Exercises""; ""Â3.2. Equivalence of Valuations; Prime Spots on a Field""; ""Â3.3. Completions, Q[sub(p)], Residue Class Fields""; ""Â3.4. Discrete Valuations""
""Â3.5. The Canonical Power Series Representation""""Â3.6. Hensel's Lemma, the Local Square Theorem, and Local Fields""; ""Â3.7. The Legendre Symbol; Recognizing Squares in Q[sub(p)]""; ""Exercises""; ""Chapter 4. Quadratic Spaces over Q[sub(p)]""; ""Â4.1. The Hilbert Symbol""; ""Â4.2. The Hasse Symbol (and an Alternative)""; ""Â4.3. Classification of Quadratic Q[sub(p)]-Spaces""; ""Â4.4. Hermitian Spaces over Quadratic Extensions of Q[sub(p)]""; ""Exercises""; ""Chapter 5. Quadratic Spaces over Q""; ""Â5.1. The Product Formula and Hilbert's Reciprocity Law""
""Â5.2. Extension of the Scalar Field""""Â5.3. Local to Global: The Hasse-Minkowski Theorem""; ""Â5.4. The Bruck-Ryser Theorem on Finite Projective Planes""; ""Â5.5. Sums of Integer Squares (First Version)""; ""Exercises""; ""Chapter 6. Lattices over Principal Ideal Domains""; ""Â6.1. Lattice Basics""; ""Â6.2. Valuations and Fractional Ideals""; ""Â6.3. Invariant factors""; ""Â6.4. Lattices on Quadratic Spaces""; ""Â6.5. Orthogonal Splitting and Triple Diagonalization""; ""Â6.6. The Dual of a Lattice""; ""Exercises""; ""Â6.7. Modular Lattices""; ""Â6.8. Maximal Lattices""
""Â6.9. Unimodular Lattices and Pythagorean Triples""""Â6.10. Remarks on Lattices over More General Rings""; ""Exercises""; ""Chapter 7. Initial Integral Results""; ""Â7.1. The Minimum of a Lattice; Definite Binary Z-Lattices""; ""Â7.2. Hermite's Bound on min L, with a Supplement for k[x]-Lattices""; ""Â7.3. Djokovic's Reduction of k[x]-Lattices; Harder's Theorem""; ""Â7.4. Finiteness of Class Numbers (The Anisotropic Case)""; ""Exercises""; ""Chapter 8. Local Classification of Lattices""; ""Â8.1. Jordan Splittings""; ""Â8.2. Nondyadic Classification""
""Â8.3. Towards 2-adic Classification""
Notes:
Description based upon print version of record.
Includes bibliographical references (pages 245-249) and index.
Description based on print version record.
ISBN:
1-4704-1159-8