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Rotatibn followed‘by reflection Richard A. Dean, About the book. . . This group oriented text in modern abstract algebra puts its emphasis on structural theorems for groups, rings, fields, and vector spaces. The motivational rationale behind each development is explained, so that the text can serve as an introduction to mathematics for anyone inter- ested in abstract algebra. The use of the important group concept brings efficiency and unity to the presentation of other algebraic systems. Constructive proofs are given whereverpossible andoften an algo- rithm suitablefor computation is ob- tained.The later chapters studyfield extensions, finite fields, Wedder- burn’s theorem, the Sylow theorems and the basis theorem for abelian groups. Galois Theory is covered in the final chapter where a full scale proofofthe criterionforapolynomial equation to be solvable in radicals is included. L.V.—8266 Elements ofAbstract Algebra Elements of Abstract Algebra RichardA. Dean California Institute ofTechnology JohnWileyandSons,Inc.,NewYork! LondonISydney Copyright© 1966byJohnWiley &Sons,Inc. AllRightsReserved Thisbookoranypartthereof mustnotbereproducedinanyform withoutthewrittenpermissionofthepublisher. LibraryofCongressCatalogCardNumber: 65—27661 PrintedintheUnited StatesofAmerica To the memory of HARVEY HarveyLewis Williams (1873—1946) taughtmathematics for41 years at Granville High School, Granville, Ohio. His counsel: “Communewith mathematics.” ' Preface This book is an outgrowth ofthe introductory algebra course given at the California Institute of Technology. The course, and more generally this textbook, is intended as an introduction to the topics and techniques of abstract algebra that are finding ever wider applications in mathematics and the applied sciences. This book also serves to introduce the student to a kind of mathematics that, in its abstraction and generality, in its applica- tion of the axiomatic method, and in its study of a proof as a means of understanding mathematics, resembles honest mathematical research more closely than does the typical course in the calculus. However, the formal prerequisites are only the equivalent ofa modern high school mathematics program. Of course, the additional experience provided by any sound course in mathematics is invaluable. InwritingthisbookIsimplywishedtosetdownmyplanforanintroductory treatment ofalgebraandto enjoytheluxury, notalways affordedin acourse, of including some topics treated infrequently at this level, and excluding other topics that have been treated extensively. In this spirit the book contains no development of linear algebra and determinants, nor is there a development of the integers from Peano postulates, nor of the real numbers from the rationals, nor of transfinite arithmetic. Whiletheseare, ofcourse, importanttopics, theyare notrequired for the material in this book. 7 ' If this book has a theme, it is the group concept. The group concept is the most central of all algebraic concepts; indeed, the techniques employed in group theory have long served as a pattern for other topics in algebra. Ifthis book has a goal, it is the Galois theoryforfields ofcharacteristic zero and a complete proof of the criterion that the zeros of a polynomial be expressable in radicals. Chapter 1 is an introduction to groups. This chapter is the longest ofthe book for it includes a number ofexamples, notably cyclic groups, permuta- tion groups, and dihedral groups. It develops the key notions of subgroup, homomorphism and factor group, and direct product. I have found that in vii viii Preface the classroom it is worth the time to treat this material in a fair amount of detail and to expect the class to be reasonably proficient in manipulating group concepts. The facility with abstraction gained here pays handsome dividends duringthe remainderofthecourse. ThusChapter2, Rings, follows easily because many ofthe theorems appear as natural analogues ofgroup- theoretic ones. In this treatment the first theorems about systems with two binary operations become routine. I have rejected the thesis that an introduction to algebra must begin with a study of the integers. While an informal discussion of number theory is interesting and useful it does not prepare a student for the subsequent abstraction of an axiomatic treatment of. groups, rings, and vector spaces. On the other hand, a development from the Peano postulates is a tedious tour de force which usually produces little useful number theory. Chapter 3, The Integers, constitutes a middle-ground approach to number theory. The integersarepresentedasan orderedintegraldomaininwhichthe ordering accomplishes a well-ordering ofthe positive elements. The fact that this axiom system is categorical is too important a result to ignore and too significant to pass offwith a “make-believe” proof. The crucial subtlety is a theorem giving sufl‘icient conditions for an inductive scheme to define a function. This section and this rather technical result may, of course, be omitted without impairing the continuity ofthe whole book. Indeed, except for the results on the arithmetic of the integers (Section 3.6), the whole chapter may be omitted, provided the reader is willing to grant the very elementary properties of ordered integral domains and well-ordered sets. Even the arithmetic is essentially repeated in the more general context of a euclidean domain in Chapter 5. The succeeding chapters on fields, euclidean domains, polynomials, and vector spaces follow in a straightforward manner. In Chapter 8, Field Extensions, the existence and uniqueness ofa splittingfield for a polynomial is theprincipalresult. The sidetripsthroughfinitefields and theWedderburn theorem on finite division rings may be included or excluded, depending on the reader’s interest. The material ofChapter 9, Finite Groups, may also be selected in this way. All ofitis essentialforarealcommand ofgroup theory, andin particularthe sections on solvablegroups areindispensablefor Galois theory. I have not yielded to the blandishments of a “slick” treatment of the classical basis theorem for finitely generated abelian groups. The proof appearing in Chapter 9 is completely constructive for finitely presented abelian groups. Throughout the book I have tried to make all proofs constructive and to give problems utilizing the constructions. The pedagogical bias of this textbook is that theory becomes meaningful when concrete problems are solved. For the uniqueness part of the basis theorem I have given P. Cohn’s proofofthe cancellation law for cyclic direct factors ofan abelian group. With this theorem a very natural proof for the uniqueness follows simply.