代数（英文版）（独家销售）

　　 本书由著名代数学家与代数几何学家Michael Artin所著，是作者在代数领域数十年的智慧和经验的结晶。书中既介绍了矩阵运算、群、向量空间、线性变换、对称等较为基本的内容，又介绍了环、模型、域，伽罗瓦理论等较为高深的内容，本书对于提高数学理解能力。增强对代数的兴趣是非常有益处的。此外，本书的可阅读性强，书中的习题也很有针对性，能让读者很快地掌握分析和思考的方法。
　　 本书在麻省理工学院、普林斯顿大学、哥伦比亚大学等著名学府得到了广泛采用，是代数学的经典教材之一。
　　

preface
a note for the teacher
chapter i matrix operations
1. the basic operations 1
2. row reduction 9
3. determinants 18
4. permutation matrices 24
5. cramer's rule 28
exercises 31
chapter 2 groups
1. the definition of a group 38
2. subgroups 44
3. isomorphisms 48
4. homomorphisms 51
$. equivalence relations and partitions 53
6. cosets 57
7. restriction of a homomorphism to a subgroup 59
8. products of groups 61
9. modular arithmetic 64
10. quotient groups 66

　　 Important though the general concepts and propositions may be with which the modern and industrious passion for axiomatizing and generalizing has presented us, in algebra perhaps more than anywhere else, nevertheless I am convinced that the special problems in all their complexity constitute the stock and core of mathematics, and that to master their difficulties requires
　　 on the whole the harder labor.
　　 Herman Weyl
　　This book began about 20 years ago in the form of supplementary notes for my algebra classes. I wanted to discuss some concrete topics such as symmetry, linear groups, and quadratic number fields in more detail than the text provided, and to shift the emphasis in group theory from permutation groups to matrix groups. Lattices, another recurring theme, appeared spontaneously. My hope was that the concrete material would interest the students and that it would make the abstractions more understandable, in short, that they could get farther by learning both at the
　　same time. This worked pretty well. It took me quite a while to decide what I wanted to put in, but I gradually handed out more notes and eventually began teaching from them without another text. This method produced a book which is, I think,somewhat different from existing ones. However, the problems I encountered while fitting the parts together caused me many headaches, so I can't recommend starting this way.
　　 The main novel feature of the book is its increased emphasis on special topics.They tended to expand each time the sections were rewritten, because I noticed over the years that, with concrete mathematics in contrast to abstract concepts, students often prefer more to less. As a result, the ones mentioned above have become major parts of the book. There are also several unusual short subjects, such as the ToddCoxeter algorithm and the simplicity of PSL2.
　　 In writing the book, I tried to follow these principles:
　　1. The main examples should precede the abstract definitions.
　　2. The book is not intended for a "service course," so technical points should be presented only if they are needled in the book.
　　3. All topics discussed should he important for the average mathematician.Though these principles may sound like motherhood and the flag, I found it useful to have them enunciated, and to keep in mind that "Do it the way you were taught"isn't one of them. They are, of course, violated here and there.
　　 The table of contents gives a good idea of the subject matter, except that a first glance may lead you to believe that the book contains all of the standard material in a beginning algebra course, and more. Looking more closely, you will find that things have been pared down here and there to make space for the special topics. I used the above principles as a guide. Thus having the main examples in hand before proceeding to the abstract material allowed some abstractions to he treated more concisely. I was also able to shorten a few discussions by deferring them until the students have already overcome their inherent conceptual difficulties. The discussion of Peano's axioms in Chapter 10, for example, has been cut to two pages. Though
　　the treatment given there is very incomplete, my experience is that it suffices to give the students the flavor of the axiomatic development of integer arithmetic. A more extensive discussion would he required if it were placed earlier in the book, and the time required for this wouldn't be well spent. Sometimes the exercise of deferring material showed that it could be deferred forever——that it was not essential. This happened with dual spaces and multilinear algebra, for example, which wound up on the floor as a consequence of the second principle. With a few concepts, such as the minimal polynomial, I ended up believing that their main purpose in introductory algebra books has been to provide a convenient source of exercises.
　　 The chapters are organized following the order in which I usually teach a course, with linear algebra, group theory, and geometry making up the first semester. Rings are first introduced in Chapter 10, though that chapter is logically independent of many earlier ones. I use this unusual arrangement because I want to emphasize the connections of algebra with geometry at the start, and because, overall, the material in the first chapters is the most important for people in other fields.
　　 The drawback is that arithmetic is given short shrift. This is made up for in the later
　　chapters, which have a strong arithmetic slant. Geometry is brought back from time
　　to time in these later chapters, in the guise of lattices, symmetry, and algebraic geometry.
　　 Michael Artin
　　 December 1990