代数（英文版）[图书]

    书籍
    数学书籍
本书系为“抽象代数”研究生课程编写的教材。
本书包含诸多代数分支主题，如有限群。环论和域论以及代数几何与模论初步等。为适应教学安排，书中题材按“非交换代数”和“交换代数”两部分组织，前一部分主要讲群论和模论基础，后一部分讲交换、域论和多项式。为便于学习，作者自然地引入基本概念和定义，并在此基础上导出相应的定理，给出严格的证明，同时举出许多例子，并在每章的结尾收入许多习题。为适应不同读者和教学内容的需求，书中还含有部分选修内容，这为那些想深入高级主题的学生提供了提高的机会，同时也便于教师按自己的风格灵活组织教学内容。

作者lsaacs将他对代数的专注和二十多年的教学经验融入到本书中，他对代数的激情与钟爱流露于字里行间。
I．Martin lsaacs是威斯康星大学麦迪逊分校数学系教授，在数学方面有丰富的教学经验，互多年教授一年级形容生代数课程。他主要在威斯康星大学麦迪逊分校任教，有时也在加州大学伯克利分校和芝加哥大学教授相应的课程。

Pt. 1 Noncommutative Algebra 1
Ch. 1 Definitions and Examples of Groups 3
Ch. 2 Subgroups and Cosets 14
Ch. 3 Homomorphisms 30
Ch. 4 Group Actions 42
Ch. 5 The Sylow Theorems and p-groups 55
Ch. 6 Permutation Groups 70
Ch. 7 New Groups from Old 83
Ch. 8 Solvable and Nilpotent Groups 99
Ch. 9 Transfer 115
Ch. 10 Operator Groups and Unique Decompositions 129
Ch. 11 Module Theory without Rings 142
Ch. 12 Rings, Ideals, and Modules 159
Ch. 13 Simple Modules and Primitive Rings 177
Ch. 14 Artinian Rings and Projective Modules 194
Ch. 15 An Introduction to Character Theory 213
Pt. 2 Commutative Algebra 231
Ch. 16 Polynomial Rings, PIDs, and UFDs 233
Ch. 17 Field Extensions 254
Ch. 18 Galois Theory 274

　　 When I started graduate school at Harvard in 1960, I knew essentially no abstract algebra. I had seen the definition of a group when I was an undergraduate, but I doubt that I had ever seen a factor group, and I am sure that I had never even heard of modules. Despite my ignorance of algebra (or perhaps because of it), I decided to register for the first half of the graduate algebra sequence. which was being taught that year by Pofessor Lynn Loomis. I found the course exciting and beautiful, and by the end of the semester I had decided that I wanted to be an algebraist. This decision was reinforced by an equally spectacular second semester.
　　 I have now been teaching mathematics for more than a quarter-century, and I have taught the two-semester first-year graduate algebra course many times. (This has been mostly at the University of Wisconsin, Madison, but I also taught parts of the corresponding courses at Chicago axld at Berkeley.) I have never forgonen Professor Loomis's course at Harvard, and in many ways. I try to imitate it. Loomis, for example, used the first semester mostly for noncommutative algebra, and he discussed commutative algebra in the second half of the course. I too divide the year this way, which is reflected in the organization of this book: Part 1 covers group theory and noncommutative rings. and Part 2 deals with field theory and commutative rings .
　　 The course that I took at Harvard "sold" me on algebra, and when I teach it. I likewise try to "sell" the subject. This affects my choice of topics, since I seldom teach a definition, for example, unless it leads to some exciting (or at least interesting) theorem. This philosophy carries over from my teaching into this book, in which I have tried to capture as well as I can the "feel" of my lectures. I would like to make my students and my readers as excited about algebra as I became during my first year of graduate school.
　　 Students in my class are expected to have had an undergraduate algebra course in which they have seen the most basic ideas of group theory, ring theory, and fieldtheory, and they are also assumed to know elementary linear algebra and matrixtheory. Most impotrant. they should be comfortable with mathematical proofs; they should know how to read them. invent them. and write them. I do not require,however, that my students actually remember the theorems or even the definitions from their undergraduate algebra courses. Given my own lack of preparation as a first-year graduate student, I am well aware that a few in my audience may be completely innocent of algebra, and I want to condnct the course so that a student such as I was can enjoy it. But this is a graduate course. and it would not be fair to the majority to go on endlessly with "review" material. I resolve this contradiction by making my presentation complete: giving all definitions and basic results, but I do this quickly. and I intersperse the review material with ideas that very few of my audience have seen before. I have attempted to do ale same in this book.
　　 By the end of a year-long graduate algebra coarse, a good student is Teady to go more deeply into one or more of the many branches of algebra. She or he might enroll in a course in finite groups, algebraic number theory. ring theory. algebraic geometry. or any of a number of other specialized topics. While I do not pretend that this book would be suitable as a text for any of these second-year courses, I have attempted to include some of the important material from many of them. I hope that this provides a convenient way for interested readers to sample a number of these topics without having to cope with the somewhat inconsistent notations and different assumptions about readers' backgrounds that are found in the various specializedmbooks. No attempt has been made, however, to designate in the text which chapters and sections are first-year material and which are second; this is simply not well defined. Lecturers who teach from this book undoubtedly do not agree on what,precisely, should be the content of a first-year course. In addition to providing opportunities for students to sample advanced topics, the additional material here should provide some fiexibility for instructors to construct a course compatible with their own tastes. Also, those with very well-prepared and capable students might
　　elect to leave out much of the "easy stuff" and build a course consisting largely of what I think of as "second-year" topics. "
　　 Since it is impossible, in my opinion, to cover all of the material in this book in a two-semester course, some topics must be skipped, and others might be assigned to the students for independent reading. Perhaps it would be useful for me to describe the content of the course as I teach it at Madison.
　　 I cover just about all of the first four chapters on basic group theory, and I do most of Chapter 5 on the Sylow theorems and p-groups, although I omit Theorem 5.27 and Section 5D on Brodkey's theorem. I do Chapter 6 on symmetric and alternating groups except for Sectept 6D. In Chapter 7, I cover direct products, but I omit Theorem 7. l 6 and Section 7C on semidirect prodacts. In Chapter 8 on solvable and nilpotent groups, I omit most of Sections 8C and 8D and all of 8F. I do present the Frattini argument (8. 10) and the most basic definitions and facts about nilpotent groups. Chapter 9 on transfer theory I skip entirely.
　　 Chapters 10 and ll on operator groups are a transition between group theory and module theory. I cover Sections 10A and 10B on the Jordan-Holder theorem for operator groups, but I omit Section 10C on the Krull-Schmidt theorem. I cover Sections 11A and 11B on chain conditions, but I touch 11C only lightly. I discuss Zorn's lemma (11.17), but I do not present a proof.
　　 Chapter 12 begins the discussion of ring theory, and some readers mav feel that there is a downward "jump discontinuity" in the level of sophistication at this point. As in Chapters l and 2. where the definitions and most basic properties of groups are presented (reviewed), it seems that here too it is imgortant to give clear definitions and discussions of elementary properties for the sake of those few readers who may not be comfortable with this material. Section 12A could be assigned as independent reading, but I usually go over it quickly in class. I cover all of Chapter 12 in my course. I do Sections 13A and 13B on the Jacobson radical completely.but sometimes I skip Section 13C on the Jacobson density theorem. (I often find that I am running short of time when I get here.) I do Sections 13D and 14A and as much of 14B on the Wedderburn-Artin theorems as I have time for, and I omit the rest of
　　 Chapter 14 and all of Chapter 15.
　　 In the second semester, I start with Chapter 16. and I cover almost all of that. except that I go lightly over Section 16D. I construct fraction fields for domains. but I do not discuss localization more generally. Chapters 17 and 18 discuss basic field extension theory and Galois theory; I cover them in their entirety. I discuss
　　 Section 19A on separability, but usually I do only a small amount of 19B on purely inseparable extensions, and I skip l9C. I cover Sections 20A and 20B on cyclotomic extensions, but I skip 20C and go very quickly over 20D on compass and straightedge constructions. In Chapter 21 on finite fields, I cover only Sections 21A and 21D: the basic material and the Wedderburn theorem on finite division rings. In Chapter 22,
　　 I omit Sections 22C and 22E, but. of course, I cover 22A and 22B on the solvability of polynomials thoroughly. and I present the fundamental theorem of algebra in 22D. In Chapter 23. I do only Sections 23A. 23B, and 23C, discussing norms and traces. Hilbert's Theorem 90, and a very rudimentary introduction to cohomology.
　　 I generally skip Chapter 24 on transcendental extensions completely. and I almost completely skip Chapter 25. (I may mention the Artin-Schreier theorem, but I never discuss formally real fields.)
　　 Chapter 26 begins the discussion of the ideal theory of commutative rings. I cover the first two sections, but I skip Section 26C on localization. In Chapterr 27 on noetherian rings. I usually cover only the first two sections and seldom get as far as 27C on the uniqueness of primes in the Lasker-Noether theorem. (I wish that
　　 I could discuss Krull's results on the heights of prime ideals in Section 27E, but it seems impossible to find the time to do that.) In Chapter 28 on integrality, I cover only the first three sections. I try to cover at least Section 29A, giving "the basic propenies of Dedekind domains, but often I find that I must skip Chapter 29 entirely because of time pressure. I always leave enough time. however, to prove Hilbert's Nullstellensatz in Section 30A, and that completes the course.
　　 The user of this book will choose what to read (or teach) and what to skip. but I, as the author, was forced to make other choices. For most of these, there were arguments in both directions. and I am certain that very few will agree with all of my decisions, and perhaps I cannot even hope for a majority agreement on each of them separately. I elected not to include tensor products. for example. because there just didn't seem to be much interesting that one could say about them without going deeply either into the theory of simple algebras or into homological algebra.
　　 Somewhat similarly, I decided not to discuss injective modules. It would have taken considerable effort just to prove that they exist in most cases, alld there did not
　　 seem much that one could do with them without going into areas of ring theory beyond what I wished to discuss. Also, I did not discuss fully the characterization of finitely generated modules over PIDs. but I did include what seem to be the two most important special cases: the fundamental theorem for finite abelian groups and the fact that torsion-free, finitely generated modules over PIDs are free.