O'Meara Introduction to Quadratic Forms

2000
ISBN: 978-3-642-62031-7
Verlag: Springer
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 344 Seiten, eBook

Reihe: Classics in Mathematics

From the reviews: "Anyone who has heard O'Meara lecture will recognize in every page of this book the crispness and lucidity of the author's style. [...] The organization and selection of material is superb. [...] deserves high praise as an excellent example of that too-rare type of mathematical exposition combining conciseness with clarity." Bulletin of the AMS

Zielgruppe

Research

Autoren/Hrsg.

O'Meara, O. Timothy

Inhaltsverzeichnis

Prerequisites ad Notation Part One: Arithmetic Theory of Fields

I Valuated Fields

Valuations

Archimedean Valuations
Non-Archimedean valuations

Prolongation of a complete valuation to a finite extension
Prolongation of any valuation to a finite separable extension

Discrete valuations

II Dedekind Theory of Ideals Dedekind axioms for S

Ideal theory

Extension fields

III Fields of Number Theory
Rational global fields
Local fields
Global fields
Part Two: Abstract Theory of Quadratic Forms
VI Quadratic Forms and the Orthogonal Group
Forms, matrices and spaces
Quadratic spaces
Special subgroups of On(V)
V The Algebras of Quadratic Forms
Tensor products
Wedderburn's theorem on central simple algebras
Extending the field of scalars
The clifford algebra
The spinor norm
Special subgroups of On(V)
Quaternion algebras
The Hasse algebra
VI The Equivalence of Quadratic Forms
Complete archimedean fields
Finite fields
Local fields
Global notation
Squares and norms in global fields
Quadratic forms over global fields
VII Hilbert's Reciprocity Law
Proof of the reciprocity law
Existence of forms with prescribed local behavior
The quadratic reciprocity law
Part Four: Arithmetic Theory of Quadratic Forms over Rings
VIII Quadratic Forms over Dedekind Domains
Abstract lattices
Lattices in quadratic spaces
IX Integral Theory of Quadratic Forms over Local Fields
Generalities

Classification of lattices over non-dyadic fields
Classification of Lattices over dyadic fields
Effective determination of the invariants
Special subgroups of On(V)
X Integral Theory of Quadratic Forms over Global Fields
Elementary properties of the orthogonal group over arithmetic fields
The genus and the spinor genus
Finiteness of class number
The class and the spinor genus in the indefinite case
The indecomposable splitting of a definite lattice
Definite unimodular lattices over the rational integers
Bibliography
Index Bibliography
Index

Über Autor(innen)

Timothy O'Meara was born on January 29, 1928. He was educated at the University of Cape Town and completed his doctoral work under Emil Artin at Princeton University in 1953. He has served on the faculties of the University of Otago, Princeton University and the University of Notre Dame. From 1978 to 1996 he was provost of the University of Notre Dame. In 1991 he was elected Fellow of the American Academy of Arts and Sciences.

O'Mearas first research interests concerned the arithmetic theory of quadratic forms. Some of his earlier work - on the integral classification of quadratic forms over local fields - was incorporated into a chapter of this, his first book.

Later research focused on the general problem of determining the isomorphisms between classical groups. In 1968 he developed a new foundation for the isomorphism theory which in the course of the next decade was used by him and others to capture all the isomorphisms among large new families of classical groups. In particular, this program advanced the isomorphism question from the classical groups over fields to the classical groups and their congruence subgroups over integral domains.

In 1975 and 1980 O'Meara returned to the arithmetic theory of quadratic forms, specifically to questions on the existence of decomposable and indecomposable quadratic forms over arithmetic domains.