Oscar Zariski Pierre Samuel Corn m utative Algebra Volume I 4 Springer Graduate Texts in Mathematics 2 8 Ediwrial Board S. Ax let FIW. Gehring P.R. }iahnos Springer I3erlin H;r1t'/oqgj lludapest Hong London \isia ri (7/gin, Oscar Zariski Pierre Samuel Commutative Algebra Volume 1 Springer Commutative Algebra VOLUME I by OSCAR ZARISKI Professor of Mat/zematks Harvard University AND PIERRE SAMUEL Professor of Mathematics University of Clermont-Ferrand WITH THE COOPERATION OF I. S. COHEN D. VAN NOSTRAND COMPANY, INC. PRINCETON, NEW JERSEY TORONTO LONDON NEW YORK D. VAN NOSTRAND COMPANY, INC. 120 Alexander St., Princeton, New Jersey (Principal office) 24 West 40 Street, New York 18, New York D. VAN NOSTRAND COMPANY, LTD. 358, Kensington High Street, London, W.14, England D. VAN NOSTRAND COMPANY (Canada), LTD. 25 Hollinger Road, Toronto 16, Canada COPYRIGHT © 1958, BY D VAN NOSTRAND COMPANY, INC. Published simultaneously in Canada by D. VAN NOSTRAND COMPANY (Canada), LTD. Library of Congress Catalogue Card No. 58-791 1 No reproduction in any form of this book, in whole or in part (except for brief quotation in critical articles or reviews), may be made without written authorization from the publishers. First Published February 1958 Reprinted June 1959, January 1962, November 1963 Reprinted February 1965 PRINTED IN THE UNITED STATES OF AMERICA PREFACE Le juge: Accuse, vous tacherez d'être bref. L'accusé: Je tâcherai d'etre clair. —G. COURTEL1NE This book is the child of an unborn parent. Some years ago the senior author began the preparation of a Colloquium volume on algebraic geom- etry, and he was then faced with the difficult task of incorporating in that volume the vast amount of purely algebraic material which is needed abstract algebraic geometry. The original plan was to insert, from time to time, algebraic digressions in which concepts and results from commu- tative algebra were to be developed in full as and when they were needed. However, it soon became apparent that such a parenthetical treatment of the purely algebraic topics, covering a wide range of commutative algebra, would impose artificial bounds on the manner, depth, and degree of gener- ality with which these topics could be treated. As is well known, abstract algebraic geometry has been recently not only the main field of applications of commutative algebra but also the principal incentive of new research in commutative algebra. To approach the underlying algebra only in a strictly utilitarian, auxiliary, and parenthetical manner, to stop short of going further afield where the applications of algebra to algebraic geometry stop and the general algebraic theories inspired by geometry begin, im- pressed us increasingly as being a program scientifically too narrow and psychologically frustrating, not to mention the distracting effect that re- peated algebraic digressions would inevitably have had on the reader, vis—à-vis the central algebro-geometric theme. Thus the idea of a separate book on commutative algebra was born, and the present book—of which this is the first of two volumes—is a realization of this idea, come to fruition at a time when its parent—a treatise on abstract algebraic geom- etry—has still to see the light of the day. In the last twenty years commutative algebra has undergone an inten- sive development. However, to the best of our knowledge, no systematic account of this subject has been published in book form since the appear- ance in 1935 of the valuable Ergebnisse monograph "Idealtheorie" of PREFACE W. Krull. As to that monograph, it has exercised a great influence on research in the intervening years, but the condensed and sketchy character of the exposition (which was due to limitation of space in the Ergebnisse monographs) made it more valuable to the expert than to the student wishing to study the subject. In the present book we endeavor to give a systematic and—we may even say—leisurely account of commutative algebra, including some of the more recent developments in this field, without pretending, however, to give an encyclopedic account of the subject matter. We have preferred to write a self-contained book which could be used in a basic graduate course of modern algebra. It is also with an eye to the student that we have tried to give full and detailed explanations in the proofs, and we feel that we owe no apology to the mature mathema- tician, who can skip the details that are not necessary for him. We have even found that the policy of trading empty space for clarity and explicit- ness of the proofs has saved us, the authors, from a number of erroneous conclusions at the more advanced stages of the book. We have also tried, this time with an eye to both the student and the mature mathematician, to give a many-sided treatment of our topics, not hesitating to offer several proofs of one and the same result when we thought that something might be learned, as to methods, from each of the proofs. The algebro-geometric origin and motivation of the book will become more evident in the second volume (which will deal with valuation theory, polynomial and power series rings, and local algebra; more will be said of that volume in its preface) than they are in this first volume. Here we develop the elements of commutative algebra which we deem to be of general and basic character. In chapter 1 we develop the introductory notions concerning groups, rings, fields, polynomial rings, and vector spaces. All this, except perhaps a somewhat detailed discussion of quotient rings with respect to multiplicative systems, is material which is usually given in an intermediate algebra course and is often briefly reviewed in the begin- ning of an advanced graduate course. The exposition of field theory given in chapter II is fairly complete and follows essentially the lines of standard modern accounts of the subject. However, as could be expected from algebraic geometers, we also stress treatment of transcendental ex- tensions, especially of the notions of separability and linear (the latter being due to A. Weil). The study of maximally algebraic subfieds and regular extensions has been postponed, however, to Volume II (chap- ter VII), since that study is so closely related to the question of ground field extension in polynomial rings. PREFACE vii Chapter III contains classical material about ideals and modules in arbitrary commutative rings. Direct sum decompositions are studied in detail. The last two sections deal respectively with tensor products of rings and free joins of integral domains. Here we introduce the notion of quasi-linear disjointness, and prove some results about free joins of inte- gral domains which we could not readily locate in the literature. With chapter IV, devoted to noetherian rings, we enter commutative algebra proper. After a preliminary section on the Hubert basis theorem and a side trip to the rings satisfying the descending chain condition, the first part of the chapter is devoted mostly to the notion of a primary repre- sentation of an ideal and to applications of that notion. We then give a detailed study of quotient rings (as generalized by Chevalley and Uzkov). The end of the chapter contains miscellaneous complements, the most im- portant of which is Krull's theory of prime ideal chains in noetherian rings. An appendix generalizes some properties of the primary representation to the case of noetherian modules. Chapter V begins with a study of integral dependence (a subject which is nowadays an essential prerequisite for almost everything in commutative algebra) and includes the so-called "going-up" and "going-down" the- orems of Cohen-Seidenberg and the normalization theorem. (Other varia- tions of that theorem will be found in Volume II, in the chapter on poly- nomial and power series rings.) With Matusita we then define a Dedekind domain as an integral domain in which every ideal is a product of prime ideals and derive from that definition the usual characterization of Dede- kind domains and their properties. An important place is given to the study of finite algebraic field extensions of the quotient field of a Dedekind domain, and the degree formula = n is derived under the usual (and necessary) finiteness assumptions concerning the integral closure of the given Dedekind domain in the extension field. This study finds its natural refinement in the Hilbert ramification theory (sections 9 and 10) and in the properties of the different and discriminant (section 11). The chap- ter closes with some classical number-theoretic applications and a generali- zation of the theorem of Kummer. The properties of Dedekind domains give us a natural opportunky of introducing the notion of a valuation (at least in the discrete case) but the reader will observe that this notion is introduced by us quite casually and parenthetically, and that the language of valuations is not used in this chapter. We have done that deliberately, for we wished to emphasize the by now well-known fact that while ideals and valuations cover substantially the same ground in the classical case (which, from a geometric point of view, is the case of dimension I), the viii PREFACE domain in which valuations become really significant belongs to the theory of function fields of dimension greater than 1. The preparation of the first volume of this book began as a collaboration between the senior author and our former pupil and friend, the late Irving S. Cohen. We extend a grateful thought to the memory of this gifted young mathematician. We wish to acknowledge many improvements in this book which are due to John Tate and Jean-Pierre Serre. We also wish to thank heartily Mr. T. Knapp who has carefully read the manuscript and the galley proofs and whose constructive criticisms have been most helpful. Thanks are also due to the Harvard Foundation for Advanced Research whose grant to the senior author was used for typing part of the manu- script. Last but not least, we wish to extend our thanks to the D. Van Nostrand Company for having generously cooperated with our wishes in the course of the printng of the book.* OsCAR ZARI5KI PIERRE SAMUEL Cambridge, Massachusetts Cliamalieres, France * The work on this volume was supported in part by a research project at Harvard University, sponsored by the Office of Ordnance Research, United States Army, under Contract DA- I 9-020-ORD-3 100. TABLE OF CONTENTS CHAPTER PAGE INTRODUCTORY CONCEPTS I. § 1. Binary operations I § 2. Groups 3 § 3. Subgroups 4 § 4. Abelian groups 6 § 5. Rings 7 § 6. Rings with identity 8 § 7. Powers and multiples 9 § 8. Fields 10 § 9. Subrings and subfields 10 § 10. Transformations and mappings 12 § II. Group homomorphisms 13 § 12. Ring homomorphisms 16 § 13. Identification of rings 19 § 14. Unique factorization domains 21 § 15. Euclidean domains 22 § 16. Polynomials in one indeterminate 24 § 17. Polynomial rings 28 § 18. Polynomials in several indeterminates 34 § 19. Quotient fields and total quotient rings 41 § 20. Quotient rings with respect to multiplicative systems 46 § 21. Vector spaces 49 II. ELEMENTS OF FIELD THEORY § 1. Field extensions 55 § 2. Algebraic quantities 55 § 3. Algebraic extensions 60 § 4. The characteristic of a field 62 § 5. Separable and inseparable algebraic extensions 65 § 6. Splitting fields and normal extensions 72 § 7. The fundamental theorem of Galois theory 80 § 8. Galois fields 82 § 9. The theorem of the primitive element 84 ix