代数 第3版（影印版）

　　 As I see it, the graduate course in algebra must primarily prepare studentsto handle the algebra which they will meet in all of mathematics: topology,partial differential equations, differential geometry, algebraic geometry, analysis,and representation theory, not to speak of algebra itself and algebraic numbertheory with all its ramifications. Hence I have inserted throughout references topapers and books which have appeared during the last decades, to indicate someof the directions in which the algebraic foundations provided by this book areused; I have accompanied these references with some motivating comments, toexplain how the topics of the present book fit into the mathematics that is tocome subsequently in various fields; and I have also mentioned some unsolvedproblems of mathematics in algebra and number theory. The abc conjecture isperhaps the most spectacular of these.
　　

part one the basic objects of algebra
chapter i groups
1. monoids 3
2. groups 7
3. normal subgroups 13
4. cyclic groups 23
5. operations of a group on a set 25
6. sylow subgroups 33
7. direct sums and free abelian groups 36
8. finitely generated abelian groups 42
9. the dual group 46
10. inverse limit and completion 49
11. categories and functors 53
12. free groups 66
chapter ii rings
1. rings and homomorphisms 83
2. commutative rings 92
3. polynomials and group rings 97
4. localization 107
5. principal and factorial rings 111

　　 The present book is meant as a basic text for a one-year course in algebra,at the graduate level.A perspective on algebra
　　 As I see it, the graduate course in algebra must primarily prepare studentsto handle the algebra which they will meet in all of mathematics: topology,partial differential equations, differential geometry, algebraic geometry, analysis,and representation theory, not to speak of algebra itself and algebraic numbertheory with all its ramifications. Hence I have inserted throughout references topapers and books which have appeared during the last decades, to indicate someof the directions in which the algebraic foundations provided by this book areused; I have accompanied these references with some motivating comments, toexplain how the topics of the present book fit into the mathematics that is tocome subsequently in various fields; and I have also mentioned some unsolvedproblems of mathematics in algebra and number theory. The abc conjecture isperhaps the most spectacular of these.
　　 Often when such comments and examples occur out of the logical order,especially with examples from other branches of mathematics, of necessity someterms may not be defined, or may be defined only later in the book. I have triedto help the reader not only by making cross-references within the book, but alsoby referring to other books or papers which I mention explicitly.
　　 I have also added a number of exercises. On the whole, I have tried to makethe exercises complement the examples, and to give them aesthetic appeal. Ihave tried to use the exercises also to drive readers toward variations and appli-cations of the main text, as well as toward working out special cases, and asopenings toward applications beyond this book.Organization
　　 Unfortunately, a book must be projected in a totally ordered way on the pageaxis, but that's not the way mathematics "is", so readers have to make choiceshow to reset certain topics in parallel for themselves, rather than in succession.I have inserted cross-references to help them do this, but different people willmake different choices at different times depending on different circumstances.
　　 The book splits naturally into several parts. The first part introduces the basicnotions of algebra. After these basic notions, the book splits in two majordirections: the direction of algebraic equations including the Galois theory inPart II; and the direction of linear and multilinear algebra in Parts III and IV.There is some sporadic feedback between them, but their unification takes placeat the next level of mathematics, which is suggested, for instance, in 15 ofChapter VI. Indeed, the study of algebraic extensions of the rationals can becarried out from two points of view which are complementary and interrelated:representing the Galois group of the algebraic closure in groups of matrices (thelinear approach), and giving an explicit determination of the irrationalities gen-erating algebraic extensions (the equations approach). At the moment, repre-sentations in GL2 are at the center of attention from various quarters, and readerswill see GL2 appear several times throughout the book. For instance, I have
　　found it appropriate to add a section describing all irreducible characters ofGL2(F) when F is a finite field. Ultimately, GL2 will appear as the simplest buttypical case of groups of Lie types, occurring both in a differential context andover finite fields or more general arithmetic rings for arithmetic applications.
　　 After almost a decade since the second edition, I find that the basic topicsof algebra have become stable, with one exception. I have added two sectionson elimination theory, complementing the existing section on the resultant.Algebraic geometry having progressed in many ways, it is now sometimes return-
　　ing to older and harder problems, such as searching for the effective constructionof polynomials vanishing on certain algebraic sets, and the older eliminationprocedures of last century serve as an introduction to those problems.
　　 Except for this addition, the main topics of the book are unchanged from thesecond edition, but I have tried to improve the book in several ways.
　　 First, some topics have been reordered. I was informed by readers and review-ers of the tension existing between having a textbook usable for relatively inex-perienced students, and a reference book where results could easily be found ina systematic arrangement. I have tried to reduce this tension by moving all thehomological algebra to a fourth part, and by integrating the commutative algebrawith the chapter on algebraic sets and elimination theory, thus giving an intro-duction to different points of view leading toward algebraic geometry.The book as a text and a reference
　　 In teaching the course, one might wish to push into the study of algebraicequations through Part II, or one may choose to go first into the linear algebraof Parts III and IV. One semester could be devoted to each, for instance. Thechapters have been so written as to allow maximal flexibility in this respect, andI have frequently committed the crime of lese-Bourbaki by repeating short argu-ments or definitions to make certain sections or chapters logically independentof each other.
　　 Granting the material which under no circumstances can be omitted from abasic course, there exist several options for leading the course in various direc-tions. It is impossible to treat all of them with the same degree of thoroughness.The precise point at which one is willing to stop in any given direction willdepend on time, place, and mood. However, any book with the aims of thepresent one must include a choice of topics, pushing ahead in deeper waters,while stopping short of full involvement.
　　 There can be no universal agreement on these matters, not even between theauthor and himself. Thus the concrete decisions as to what to include and whatnot to include are finally taken on grounds of general coherence and aestheticbalance. Anyone teaching the course will want to impress their own personalityon the material, and may push certain topics with more vigor than I have, at theexpense of others. Nothing in the present book is meant to inhibit this.
　　 Unfortunately, the goal to present a fairly comprehensive perspective onalgebra required a substantial increase in size from the first to the second edition,and a moderate increase in this third edition. These increases require somedecisions as to what to omit in a given course.
　　 Many shortcuts can be taken in the presentation of the topics, whichadmits many variations. For instance, one can proceed into field theory andGalois theory immediately after giving the basic definitions for groups, rings,fields, polynomials in one variable, and vector spaces. Since the Galois theorygives very quickly an impression of depth, this is very satisfactory in manyrespects.
　　 It is appropriate here to recall my original indebtedness to Artin, who firsttaught me algebra. The treatment of the basics of Galois theory is muchinfluenced by the presentation in his own monograph.
　　Audience and background
　　 As I already stated in the forewords of previous editions, the present bookis meant for the graduate level, and I expect most of those coming to it to havehad suitable exposure to some algebra in an undergraduate course, or to haveappropriate mathematical maturity. I expect students taking a graduate courseto have had some exposure to vector spaces, linear maps, matrices, and theywill no doubt have seen polynomials at the very least in calculus courses.
　　 My books Undergraduate Algebra and Linear Algebra provide more thanenough background for a graduate course. Such elementary texts bring out inparallel the two basic aspects of algebra, and are organized differently from thepresent book, where both aspects are deepened. Of course, some aspects of thelinear algebra in Part III of the present book are more "elementary" than someaspects of Part II, which deals with Galois theory and the theory of polynomialequations in several variables. Because Part II has gone deeper into the studyof algebraic equations, of necessity the parallel linear algebra occurs only laterin the total ordering of the book. Readers should view both parts as runningsimultaneously.