This book examines algebraic number theory and the theory of semisimple algebras. It covers classification over an algebraic number field and classification over the ring of algebraic integers.

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.

Discusses algebraic number theory and the theory of semisimple algebras Discusses classification of quadratic forms over the ring of algebraic integers Discusses local class field theory Presents a new framework of quadratic Diophantine equations Includes supplementary material: sn.pub/extras

Autorentext

Goro Shimura is currently a professor emeritus of mathematics at Princeton University.

Inhalt
The Quadratic Reciprocity Law.- Arithmetic in an Algebraic Number Field.- Various Basic Theorems.- Algebras Over a Field.- Quadratic Forms.- Deeper Arithmetic of Quadratic Forms.- Quadratic Diophantine Equations.