Sharpe & Vamos, Infective Modules (Cambridge Tract in Mathematics, Cambridge University Press, 1972), 143 pp., £5·00.
P. F. Smith · Proceedings of the Edinburgh Mathematical Society · 1974
This book was based on a graduate course given at the University of Rochester during 1969-70.Chapter I contains a survey of the standard work on absolute values (archimedian and non-archimedian) normally given for applications in algebraic number theory.Adequate references are given for proofs of the results that are merely stated.Chapter II is entitled " Valuation Rings " and generalises valuations to Krull valuations.The idea of a " place " is introduced and the connections between places and valuation rings developed.The chapter introduces usual terms such as integrally closed rings, Priifer rings, Krull rings and Dedekind rings, discusses their interrelations, and describes a variety of results like the Chinese remainder theorem and the approximation theorems.In Chapter III, the main chapter of the book, the important subject of extensions of valuation rings is covered.This involves the usual work on algebraic and normal field extensions, gives a clear presentation of henselian valuation rings and related work, and contains infinite ramification theory.In the final chapter, Chapter IV, the author includes work of his own in discussing fields with prescribed valuations; the work admits archimedian valuations and generalises results of Krull.At the end of the book there is an interesting set of exercises listed under the four chapter headings and a useful bibliography.Like all books based directly on a set of lectures and seminars it reflects to some extent the personal interests of the author.However the selection and presentation of material are excellent and the book is a welcome addition to the literature on valuation theory.