Arithmetic of quadratic forms

✍ Scribed by Goro Shimura (auth.)

• Publisher: Springer-Verlag New York
• Year: 2010
• Tongue: English
• Leaves: 250
• Series: Springer Monographs in Mathematics
• Edition: 1
• Category: Library

✦ Synopsis

This book is divided into two parts. The first part is preliminary and consists of algebraic number theory and the theory of semisimple algebras. There are two principal topics: classification of quadratic forms and quadratic Diophantine equations. The second topic is a new framework which contains the investigation of Gauss on the sums of three squares as a special case. To make the book concise, the author proves some basic theorems in number theory only in some special cases. However, the book is self-contained when the base field is the rational number field, and the main theorems are stated with an arbitrary number field as the base field. So the reader familiar with class field theory will be able to learn the arithmetic theory of quadratic forms with no further references.

✦ Table of Contents

Front Matter....Pages i-xi
The Quadratic Reciprocity Law....Pages 1-14
Arithmetic in an Algebraic Number Field....Pages 15-45
Various Basic Theorems....Pages 47-78
Algebras Over a Field....Pages 79-114
Quadratic Forms....Pages 115-151
Deeper Arithmetic of Quadratic Forms....Pages 153-202
Quadratic Diophantine Equations....Pages 203-232
Back Matter....Pages 233-237

✦ Subjects

Algebra; Number Theory; General Algebraic Systems