### 基本信息

- 原書名：Algebra (2nd Edition)
- 原出版社： Addison Wesley

- 作者：
**(美)Michael Artin** - 叢書名：
**華章數學原版精品系列** - 出版社：機械工業出版社
- ISBN：
**9787111367017** - 上架時間：2012-2-9
- 出版日期：2012 年1月
- 開本：16開
- 頁碼：543
- 版次：2-1
- 所屬分類：數學 > 代數，數論及組合理論 > 綜合

教材

### 編輯推薦

是代數學的經典教材之一

著名代數學家與代數幾何學家michael artin所著

在麻省理工學院、普林斯頓大學、哥倫比亞大學等著名學府得到了廣泛采用

### 內容簡介

數學書籍

《代數(英文版.第2版)》由著名代數學家與代數幾何學家Michael Artin所著，是作者在代數領域數十年的智慧和經驗的結晶。書中既介紹了矩陣運算、群、向量空間、線性算子、對稱等較為基本的內容，又介紹了環、模型、域、伽羅瓦理論等較為高深的內容。本書對于提高數學理解能力，增強對代數的興趣是非常有益處的。此外，本書的可閱讀性強，書中的習題也很有針對性，能讓讀者很快地掌握分析和思考的方法。

作者結合這20年來的教學經歷及讀者的反饋，對本版進行了全面更新，更強調對稱性、線性群、二次數域和格等具體主題。本版的具體更新情況如下：

新增球面、乘積環和因式分解的計算方法等內容，并補充給出一些結論的證明，如交錯群是簡單的、柯西定理、分裂定理等。

修訂了對對應定理、SU2 表示、正交關系等內容的討論，并把線性變換和因子分解都拆分為兩章來介紹。

新增大量習題，并用星號標注出具有挑戰性的習題。

《代數(英文版.第2版)》在麻省理工學院、普林斯頓大學、哥倫比亞大學等著名學府得到了廣泛采用，是代數學的經典教材之一。

### 作譯者

### 目錄

Preface

1 Matrices

1.1 The Basic Operations

1.2 Row Reduction

1.3 The Matrix Transpose

1.4 Determinants

1.5 Permutations

1.6 Other Formulas for the Determinant

Exercises

2 Groups

2.1 Laws of Composition

2.2 Groups and Subgroups

2.3 Subgroups of the Additive Group of Integers.

2.4 Cyclic Groups

2.5 Homomorphisms

2.6 Isomorphisms

2.7 Equivalence Relations and Partitions

2.8 Cosets

2.9 Modular Arithmetic

### 前言

--Herman Weyl

This book began many 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 to include, but I gradually handed out more notes and eventually began teaching from them without another text. Though this produced a book that is different from most others, the problems I encountered while fitting the parts together caused me many headaches. I can't recommend the method.

There is more emphasis on special topics here than in most algebra books. They tended to expand when the sections were rewritten, because I noticed over the years that, in contrast to abstract concepts, with concrete mathematics students often prefer more to less. As a result, the topics mentioned above have become major parts of the book.

In writing the book, I tried to follow these principles:

1. The basic examples should precede the abstract definitions.

2. Technical points should be presented only if they are used elsewhere in the book.

3. All topics should be important for the average mathematician.

Although these principles may sound like motherhood and the flag, I found it useful to have them stated explicitly. They are, of course, violated here and there.

The chapters are organized in 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 11, though that chapter is logically independent of many earlier ones. I chose this arrangement 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 first half of the book doesn't emphasize arithmetic, but this is made up for in the later chapters.

About This Second Edition

The text has been rewritten extensively, incorporating suggestions by many people as well as the experience of teaching from it for 20 years. I have distributed revised sections to my class all along, and for the past two years the preliminary versions have been used as texts. As a result, I've received many valuable suggestions from the students. The overall organization of the book remains unchanged, though I did split two chapters that seemed long.

There are a few new items. None are lengthy, and they are balanced by cuts made elsewhere. Some of the new items are an early presentation of Jordan form (Chapter 4), a short section on continuity arguments (Chapter 5), a proof that the alternating groups are simple (Chapter 7), short discussions of spheres (Chapter 9), product rings (Chapter 11), computer methods for factoring polynomials and Cauchy's Theorem bounding the roots of a polynomial (Chapter 12), and a proof of the Splitting Theorem based on symmetric functions (Chapter 16). I've also added a number of nice exercises. But the book is long enough, so I've tried to resist the temptation to add material.

NOTES FOR THE TEACHER

This book is designed to allow you to choose among the topics. Don't try to cover the book, but do include some of the interesting special topics such as symmetry of plane figures, the geometry of SU2, or the arithmetic of imaginary quadratic number fields. If you don't want to discuss such things in your course, then this is not the book for you.

There are relatively few prerequisites. Students should be familiar with calculus, the basic properties of the complex numbers, and mathematical induction. An acquaintance with proofs is obviously useful. The concepts from topology that are used in Chapter 9, Linear Groups, should not be regarded as prerequisites.

I recommend that you pay attention to concrete examples, especially throughout the early chapters. This is very important for the students who come tO the course without a clear idea of what constitutes a proof.

One could spend an entire semester on the first five chapters, but since the real fun starts with symmetry in Chapter 6, that would defeat the purpose of the book. Try to get to Chapter 6 as soon as possible, so that it can be done at a leisurely pace. In spite of its immediate appeal, symmetry isn't an easy topic. It is easy to be carried away and leave the students behind.

These days most of the students in my classes are familiar with matrix operations and modular arithmetic when they arrive. I've not been discussing the first chapter on matrices in class, though I do assign problems from that chapter. Here are some suggestions for Chapter 2, Groups.