This book can be divided into two parts. The ?rst part is preliminary
and consists of algebraic number theory and the theory of semisimple
algebras. The raison d'^ etre of the book is in the second part, and so
let us ?rst explain the contents of the second part. There are two
principal topics: (A) Classi?cation of quadratic forms; (B) Quadratic
Diophantine equations. Topic (A) can be further divided into two types
of theories: (a1) Classi?cation over an algebraic number ?eld; (a2)
Classi?cation over the ring of algebraic integers. To classify a
quadratic form ? over an algebraic number ?eld F, almost all previous
authors followed the methods of Helmut Hasse. Namely, one ?rst takes ?
in the diagonal form and associates an invariant to it at each prime
spot of F, using the diagonal entries. A superior method was introduced
by Martin Eichler in 1952, but strangely it was almost completely
ignored, until I resurrected it in one of my recent papers. We associate
an invariant to ? at each prime spot, which is the same as Eichler's,
but we de?ne it in a di?erent and more direct way, using Cli?ord
algebras. In Sections 27 and 28 we give an exposition of this theory. At
some point we need the Hasse norm theorem for a quadratic extension of a
number ?eld, which is included in class ?eld theory. We prove it when
the base ?eld is the rational number ?eld to make the book
self-contained in that case.
