Part I of this monograph, consisting of the first five chapters, is purely algebraic and has as its object the classification of quadratic forms over the rational field. To this end, the first three chapters deal with finite and $p$-adic fields and Hilbert's symbol. The fourth chapter accomplishes the transition from $p$-adic fields to the rational field culminating in the Hasse-Minkowski theorem and an application to the sum of three squares. The fifth chapter deals with integral quadratic forms of discriminant $±1$.

Part II is analytic in nature, Chapter VI giving the proof of Dirichlet's theorem on primes in an arithmetic progression and Chapter VII dealing with modular forms including Hecke's operators and a brief treatment of theta functions.

These two parts correspond to courses given in 1962 and 1964 to second year students in l'École Normale Supérieure. While not a text in the American sense since is has no exercises, it is very readable in spite of moving quite quickly into rather deep mathematics. Of great help is the index of symbols and terminology at the end. This is a notable contribution to the literature on quadratic forms.

Reviewed by Burton W. Jones