Table Of Content

73 Graduate Texts in Mathematics Editorial Board S. Axler F.W. Gehring K.A. Ribet Springer New York Berlin Heidelberg Hong Kong London Milan Paris Tokyo Graduate Texts in Mathematics TAKEUTI/ZARING. Introduction to 36 KELLEy/NAMIOKA et al. Linear Axiomatic Set Theory. 2nd ed. Topological Spaces. 2 OXTOBY. Measure and Category. 2nd ed. 37 MONK. Mathematical Logic. 3 SCHAEFER. Topological Vector Spaces. 38 GRAUERT/FRlTZSCHE. Several Complex 4 HILTON/STAMMBACH. A Course in Variables. Homological Algebra. 2nd ed. 39 ARVESON. An Invitation to C*-Algebras. 5 MAc LANE. Categories for the Working 40 KEMENY/SNELL/KNAPP. Denumerable Mathematician. 2nd ed. Markov Chains. 2nd ed. 6 HUGHES/PIPER. Projective Planes. 41 APOSTOL. Modular Functions and 7 SERRE. A Course in Arithmetic. Dirichlet Series in Number Theory. 8 TAKEUTI/ZARING. Axiomatic Set Theory. 2nd ed. 9 HUMPHREYS. Introduction to Lie Algebras 42 SERRE. Linear Representations of Finite and Representation Theory. Groups. 10 COHEN. A Course in Simple Homotopy 43 GILLMAN/JERISON. Rings of Continuous Theory. Functions. il CONWAY. Functions of One Complex 44 KENDIG. Elementary Algebraic Geometry. Variable I. 2nd ed. 45 LOEWE. Probability Theory I. 4th ed. 12 BEALS. Advanced Mathematical Analysis. 46 LOEVE. Probability Theory II. 4th ed. 13 ANDERSON/FULLER. Rings and Categories 47 MorSE. Geometric Topology in of Modules. 2nd ed. Dimensions 2 and 3. 14 GOLUBITSKy/GUILLEMIN. Stable Mappings 48 SACHSlWu. General Relativity for and Their Singularities. Mathematicians. 15 BERBERIAN. Lectures in Functional 49 GRUENBERGIWEIR. Linear Geometry. Analysis and Operator Theory. 2nd ed. 16 WINTER. The Structure of Fields. 50 EDWARDS. Fermat's Last Theorem. 17 ROSENBLATT. Random Processes. 2nd ed. 51 KLINGENBERG. A Course in Differential 18 HALMOS. Measure Theory. Geometry. 19 HALMOS. A Hilbert Space Problem Book. 52 HARTSHORNE. Algebraic Geometry. 2nd ed. 53 MANIN. A Course in Mathematical Logic, 20 HUSEMOLLER, Fibre Bundles. 3rd ed. 54 GRAVERIW ATKINS, Combinatorics with 21 HUMPHREYS, Linear Algebraic Groups, Emphasis on the Theory of Graphs. 22 BARNEsIMACK, An Algebraic Introduction 55 BROWN/PEARCY, Introduction to Operator to Mathematical Logic, Theory I: Elements of Functional 23 GREUB, Linear Algebra, 4th ed, Analysis, 24 HOLMES, Geometric Functional Analysis 56 MASSEY, Algebraic Topology: An and Its Applications, Introduction. 25 HEWITT/STROMBERG. Real and Abstract 57 CROWELL/Fox, Introduction to Knot Analysis. Theory, 26 MANES, Algebraic Theories, 58 KOBLITZ. p-adic Numbers, p-adic 27 KELLEY, General Topology. Analysis, and Zeta-Functions, 2nd ed, 28 ZARISKI/SAMUEL. Commutative Algebra, 59 LANG, Cyclotomic Fields. VoLI. 60 ARNOLD, Mathematical Methods in 29 ZARISKI/SAMUEL. Commutative Algebra, Classical Mechanics, 2nd ed. VoLll. 61 WHITEHEAD, Elements of Homotopy 30 JACOBSON, Lectures in Abstract Algebra I. Theory. Ba~ic Concepts, 62 KARGAPoLOvIMERLZJAKOV, Fundamentals 31 JACOBSON, Lectures in Abstract Algebra of the Theory of Groups, II. Linear Algebra, 63 BOLLOBAS, Graph Theory, 32 JACOBSON. Lectures in Abstract Algebra 64 EDWARDS, Fourier Series. Vol. I 2nd ed, Ill. Theory of Fields and Galois Theory. 65 WELLS, Differential Analysis on Complex 33 HIRSCH, Differential Topology, Manifolds. 2nd ed. 34 SPITZER, Principles of Random Walk, 66 WATERHOUSE, Introduction to Affine 2nd ed, Group Schemes, 35 ALEXANDERIWERMER. Several Complex 67 SERRE. Local Fields, Variables and Banach Algebras, 3rd ed. (continued after index) Thomas W. Hungerford ALGEBRA , Springer Thomas W. Hungerford Department of Mathematics Cleveland State University Cleveland, OH 44115 USA Editorial Board S. Axler F.W. Gehring K.A. Ribet Mathematics Department Mathematics Department Mathematics Department San Francisco State East Hall University of California, University University of Michigan Berkeley San Francisco, CA 94132 Ann Arbor, MI 48\09 Berkeley, CA 94720-3840 USA USA USA [email protected] fgehring@ [email protected] math.lsa. umich.edu Mathematics Subject Classification (2000): 26-01 Library of Congress Cataloging-in-Publication Data Hungerford, Thomas W. Algebra Bibliography: p. 1. Algebra 1. Title QA155.H83 512 73-15693 ISBN-13: 978-1-4612-6103-2 e-ISBN-13: 978-1-4612-6101-8 DOl: 10.1007/978-1-4612-6101-8 © 1974 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 15t edition 1974 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA); except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. (ASC/SBA) 15 14 13 SPIN 11013129 Springer-Verlag is a part of Springer Science+Business Media springeronline.com To Mary Preface to the Springer Edition The reception given to the first edition of Algebra indicates that is has filled a definite need: to provide a self-contained, one-volume, graduate level algebra text that is readable by the average graduate student and flexible enough to accomodate a wide variety of instructors and course contents. Since it has been so well re ceived, an extensive revision at this time does not seem warranted. Therefore, no substantial changes have been made in the text for this revised printing. How ever, all known misprints and errors have been corrected and several proofs have been rewritten. I am grateful to Paul Halmos and F. W. Gehring, and the Springer staff, for their encouragement and assistance in bringing out this edition. It is gratifying to know that Algebra will continue to be available to the mathematical community. Springer-Verlag is to be commended for its willingness to continue to produce high quality mathematics texts at a time when many other publishers are looking to less elegant but more lucrative ventures. Seattle, Washington THOMAS W. HUNGERFORD June, 1980 Note on the twelfth printing (2003): A number of corrections were incorporated in the fifth printing, thanks to the sharp-eyed diligence of George Bergman and his students at Berkeley and Keqin Feng of the Chinese University of Science and Technology. Additional corrections appear in this printing, thanks to Victor Boyko, Bob Cacioppo, Joe L. Mott, Robert Joly, and Joe Brody. vii Preface Note: A complete discussion of possible ways of using this text, including sug gested course outlines, is given on page xv. This book is intended to serve as a basic text for an algebra course at the beginning graduate level. Its writing was begun several years ago when I was unable to find a one-volume text which I considered suitable for such a course. My criteria for "suitability," which I hope are met in the present book, are as follows. (i) A conscious effort has been made to produce a text which an average (but reasonably prepared) graduate student might read by himself without undue diffi culty. The stress is on clarity rather than brevity. (ii) For the reader's convenience the book is essentially self-contained. Con sequently it includes much undergraduate level material which may be easily omitted by the better prepared reader. (iii) Since there is no universal agreement on the content of a first year graduate algebra course we have included more material than could reasonably be covered in a single year. The major areas covered are treated in sufficient breadth and depth for the first year graduate level. Unfortunately reasons of space and economics ha ve forced the omission of certain topics, such as valuation theory. For the most part these omitted subjects are those which seem to be least likely to be covered in a one year course. (iv) The text is arranged to provide the instructor with maximum flexibility in the choice, order and degree of coverage of topics. without sacrificing readability for the student. (v) There is an unusually large number of exercises. There are, in theory, no formal prerequisites other than some elementary facts about sets, functions, the integers, and the real numbers, and a certain amount of "mathematical maturity." In actual practice, however, an undergraduate course in modern algebra is probably a necessity for most students. Indeed the book is written on this assumption, so that a number of concepts with which the typical graduate student may be assumed to be acquainted (for example, matrices) are presented in examples, exercises, and occasional proofs before they are formally treated in the text. ix x PREFACE The guiding philosophical principle throughout the book is that the material should be presented in the maximum useable generality consistent with good pedago gy. The principle is relatively easy to apply to various technical questions. It is more difficult to apply to broader questions of conceptual organization. On the one hand, for example, the student must be made aware of relatively recent insights into the nature of algebra: the heart of the matter is the study of morphisms (maps); many deep and important concepts are best viewed as universal mapping properties. On the other hand, a high level of abstraction and generality is best appreciated and fully understood only by those who have a firm grounding in the special situations which motivated these abstractions. Consequently, concepts which can be character ized by a universal mapping property are not defined via this property if there is available a definition which is more familiar to or comprehensible by the student. In such cases the universal mapping property is then given in a theorem. Categories are introduced early and some terminology of category theory is used frequently thereafter. However, the language of categories is employed chiefly as a useful convenience. A reader who is unfamiliar with categories should have little difficulty reading most of the book, even as a casual reference. Nevertheless, an instructor who so desires may give a substantial categorical flavor to the entire course without difficulty by treating Chapter X (Categories) at an early stage. Since it is essentially independent of the rest of the book it may be read at any time. Other features of the mathematical exposition are as follows. Infinite sets, infinite cardinal numbers, and transfinite arguments are used routine ly. All of the necessary set theoretic prerequisites, including complete proofs of the relevant facts of cardinal arithmetic, are given in the Introduction. The proof of the Sylow Theorems suggested by R. J. Nunke seems to clarify an area which is frequently confusing to many students. Our treatment of Galois theory is based on that of Irving Kaplansky, who has successfully extended certain ideas of Emil Artin. The Galois group and the basic connection between subgroups and subfields are defined in the context of an ab solutely general pair of fields. Among other things this permits easy generalization of various results to the infinite dimensional case. The Fundamental Theorem is proved at the beginning, before splitting fields, normality, separability, etc. have been introduced. Consequently the very real danger in many presentations, namely that student will lose sight of the forest for the trees, is minimized and perhaps avoided entirely. In dealing with separable field extensions we distinguish the algebraic and the transcendental cases. This seems to be far better from a pedogogical standpoint than the Bourbaki method of presenting both cases simultaneously. If one assumes that all rings have identities, all homomorphisms preserve identi ties and all modules are unitary, then a very quick treatment of semisimple rings and modules is possible. Unfortunately such an approach does not adequately pre pare a student to read much of the literature in the theory of noncommutative rings. Consequently the structure theory of rings (in particular, semisimple left Artinian rings) is presented in a more general context. This treatmen.t includes the situation mentioned above, but also deals fully with rings without identity, the Jacobson radical and related topics. In addition the prime radical and Goldie's Theorem on semiprime rings are discussed. There are a large number of exercises of varying scope and difficulty. My experi ence in attempting to "star" the more difficult ones has thoroughly convinced me of PREFACE xi the truth of the old adage: one man's meat is another's poison. Consequently no exercises are starred. The exercises are important in that a student is unlikely to appreciate or to master the material fully if he does not do a reasonable number of exercises. But the exercises are not an integral part of the text in the sense that non trivial proofs of certain needed results are left entirely to the reader as exercises. Nevertheless, most students are quite capable of proving nontrivial propositions provided that they are given appropriate guidance. Consequently, some theorems in the text are followed by a "sketch of proor' rather than a complete proof. Some times such a sketch is no more than a reference to appropriate theorems. On other occasions it may present the more difficult parts of a proof or a necessary "trick" in full detail and omit the rest. Frequently all the major steps of a proof will be stated, with the reasons or the routine calculational details left to the reader. Some of these latter "sketches" would be considered complete proofs by many people. In such cases the word "sketch" serves to warn the student that the proof in question is somewhat more concise than and possibly not as easy to follow as some of the "complete" proofs given elsewhere in the text. Seattle, Washington THOMAS w. HUNGERFORD September, 1973