Introduction to Quadratic Forms

Timothy O'Meara was born on January 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.

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

"The exposition follows the tradition of the lectures of Emil Artin who enjoyed developing a subject from first principles and devoted much research to finding the simplest proofs at every stage." - American Mathematical Monthly

"The exposition follows the tradition of the lectures of Emil Artin who enjoyed developing a subject from first principles and devoted much research to finding the simplest proofs at every stage." - American Mathematical Monthly