Introduction to Quadratic Forms

1. Front Matter

   Pages I-XIII

   PDF

2. Arithmetic Theory of Fields

   1. Valuated Fields

      • O. Timothy O’Meara

      Pages 1-41

   2. Dedekind Theory of Ideals

      • O. Timothy O’Meara

      Pages 41-54

   3. Fields of Number Theory

      • O. Timothy O’Meara

      Pages 54-81

3. Abstract Theory Quadratic Forms

   1. Quadratic Forms and the Orthogonal Group

      • O. Timothy O’Meara

      Pages 82-112

   2. The Algebras of Quadratic Forms

      • O. Timothy O’Meara

      Pages 112-153

4. Arithmetic Theory of Quadratic Forms over Fields

   1. The Equivalence of Quadratic Forms

      • O. Timothy O’Meara

      Pages 154-189

   2. Hilbert’s Reciprocity Law

      • O. Timothy O’Meara

      Pages 190-207

5. Abstract Theory of Quadratic Forms over Rings

   1. Quadratic Forms over Dedekind Domains

      • O. Timothy O’Meara

      Pages 208-239

   2. Integral Theory of Quadratic Forms over Local Fields

      • O. Timothy O’Meara

      Pages 239-284

   3. Integral Theory of Quadratic Forms over Global Fields

      • O. Timothy O’Meara

      Pages 284-335

6. Back Matter

   Pages 336-342

   PDF

From the reviews: "O'Meara treats his subject from this point of view (of the interaction with algebraic groups). He does not attempt an encyclopedic coverage ...nor does he strive to take the reader to the frontiers of knowledge... . Instead he has given a clear account from first principles and his book is a useful introduction to the modern viewpoint and literature. In fact it presupposes only undergraduate algebra (up to Galois theory inclusive)... The book is lucidly written and can be warmly recommended.
J.W.S. Cassels, The Mathematical Gazette, 1965
"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...
R. Jacobowitz, Bulletin of the AMS, 1965

• Arithmetic Theory of Fields
• Arithmetic Theory of Rings
• Lattice
• Quadratic Forms
• algebra
• finite field
• number theory
• matrix theory

"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

• Department of Mathematics, University of Notre Dame, Notre Dame, USA

  O. Timothy O’Meara

Biography of  O. Timothy O'Meara

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.

Policies and ethics