域论(第2版)(英文版)
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- 原书名： Field Theory Second Edition
- 原出版社： Springer
- 作者： Steven Roman
- 出版社：世界图书出版公司
- ISBN：9787510037634
- 上架时间：2011-10-31
- 出版日期：2011 年7月
- 开本：24开
- 页码：332
- 版次：2-1
- 所属分类：
数学 > 初等数学
编辑推荐
一部研究生水平的域论的入门书籍
内容简介回到顶部↑
《域论(第2版)(英文版)》是一部研究生水平的域论的入门书籍。每节后面都有不少练习，使得本书既是一本很好的教程，也是一本不错的参考书。本书从头开始阐述了域基本理论，如果具备本科生水平的抽象代数知识将对学习本书具有很大的帮助。本书是第二版，作者基于第一版及在运用第一版在教学过程中的经验，又将本书中的基本内容进行了改进。增加了新的练习和新的一章从历史展望角度讲述了 Galois理论，通书不断涌现新话题，包括代数基本理论的证明、不可约情形的讨论、Zp上多项式因式分解的Berlekamp代数等。目次：基础；（第一部分）域扩展：多项式；域扩展；嵌入和可分性；代数独立性；（第二部分）Galois理论Ⅰ，历史回顾；Galois理论Ⅱ，理论；Galois理论Ⅲ，多项式的Galois群；域扩展作为向量空间；有限域Ⅰ，基本性质；有限域Ⅱ，附加性质；单位根；循环扩张；可解性扩张；（第三部分）二项式；二项式族。
目录回到顶部↑
《域论(第2版)(英文版)》
preface
contents
0 preliminaries
0.1 lattices
0.2 groups
0.3 the symmetric group
0.4 rings
0.5 integral domains
0.6 unique factorization domains
0.7 principal ideal domains
0.8 euclidean domains
0.9 tensor products
exercises
part i-field extensions
1 polynomials
1.1 polynomials over a ring
1.2 primitive polynomials and irreducibility
1.3 the division algorithm and its consequences
1.4 splitting fields
preface
contents
0 preliminaries
0.1 lattices
0.2 groups
0.3 the symmetric group
0.4 rings
0.5 integral domains
0.6 unique factorization domains
0.7 principal ideal domains
0.8 euclidean domains
0.9 tensor products
exercises
part i-field extensions
1 polynomials
1.1 polynomials over a ring
1.2 primitive polynomials and irreducibility
1.3 the division algorithm and its consequences
1.4 splitting fields
前言回到顶部↑
This book presents the basic theory of fields, starting more or less from the beginning. It is suitable for a graduate course in field theory, or independent study. The reader is expected to have taken an undergraduate course in abstract algebra, not so much for the material it contains but in order to gain a certain level of mathematical maturity.
The book begins with a preliminary chapter (Chapter 0), which is designed to be quickly scanned or skipped and used as a reference if needed. The remainder of the book is divided into three parts.
Part 1, entitled Field Extensions, begins with a chapter on polynomials. Chapter 2 is devoted to various types of field extensions, including finite, finitely generated, algebraic and normal. Chapter 3 takes a close look at the issue of separability. In my classes, I generally cover only Sections 3.1 to 3.4 (on perfect fields). Chapter 4 is devoted to algebraic independence, starting with the general notion of a dependence relation and concluding with Luroth's theorem on intermediate fields of a simple transcendental extension.
Part 2 of the book is entitled Galois Theory. Chapter 5 examines Galois theory from an historical perspective, discussing the contributions from Lagrange,Vandermonde, Gauss, Newton, and others that led to the development of the theory. I have also included a very brief look at the very brief life of Galois himself.
Chapter 6 begins with the notion of a Galois correspondence between two partially ordered sets, and then specializes to the Galois correspondence of a field extension, concluding with a brief discussion of the Krull topology. In Chapter 7, we discuss the Galois theory of equations. In Chapter 8, we view a field extension E of F as a vector space over F.
Chapter 9 and Chapter 10 are devoted to finite fields, although this material can be omitted in order to reach the topic of solvability by radicals more quickly.Mobius inversion is used in a few places, so an appendix has been included on this subject.
Part 3 of the book is entitled The Theory of Binomials. Chapter 11 covers the roots of unity and Wedderbum's theorem on finite division rings. We also briefly discuss the question of whether a given group is the Galois group of a field extension. In Chapter 12, we characterize cyclic extensions and splitting fields of binomials when the base field contains appropriate roots of unity.Chapter 13 is devoted to the question of solvability of a polynomial equation by radicals. (This chapter might make a convenient ending place in a graduate course.) In Chapter 14, we determine conditions that characterize the irreducibility of a binomial and describe the Galois group of a binomial. Chapter 15 briefly describes the theory of families of binomials--the so-called Kummer theory.
Sections marked with an asterisk may be skipped without loss of continuity. hanges for the Second Edition
Let me begin by thanking the readers of the first edition for their many helpful comments and suggestions.
For the second edition, I have gone over the entire book, and rewritten most of it, including the exercises. I believe the book has benefited significantly from a class testing at the beginning graduate level and at a more advanced graduate level.
I have also rearranged the chapters on separability and algebraic independence,feeling that the former is more important when time is of the essence. In my course, I generally touch only very lightly (or skip altogether) the chapter on algebraic independence, simply because of time constraints.
As mentioned earlier, as several readers have requested, 1 have added a chapter on Galois theory from an historical perspective.
A few additional topics are sprinkled throughout, such as a proof of the Fundamental Theorem of Algebra, a discussion of casus irreducibilis,Berlekamp's algorithm for factoring polynomials over Zp and natural and accessory irrationalities.
Thanks
I would like to thank my students Phong Le, Sunil Chetty, Timothy Choi and Josh Chan, who attended lectures on essentially the entire book and offeredmany helpful suggestions. I would also like to thank my editor, Mark Spencer,who puts up with my many requests and is most amiable.
The book begins with a preliminary chapter (Chapter 0), which is designed to be quickly scanned or skipped and used as a reference if needed. The remainder of the book is divided into three parts.
Part 1, entitled Field Extensions, begins with a chapter on polynomials. Chapter 2 is devoted to various types of field extensions, including finite, finitely generated, algebraic and normal. Chapter 3 takes a close look at the issue of separability. In my classes, I generally cover only Sections 3.1 to 3.4 (on perfect fields). Chapter 4 is devoted to algebraic independence, starting with the general notion of a dependence relation and concluding with Luroth's theorem on intermediate fields of a simple transcendental extension.
Part 2 of the book is entitled Galois Theory. Chapter 5 examines Galois theory from an historical perspective, discussing the contributions from Lagrange,Vandermonde, Gauss, Newton, and others that led to the development of the theory. I have also included a very brief look at the very brief life of Galois himself.
Chapter 6 begins with the notion of a Galois correspondence between two partially ordered sets, and then specializes to the Galois correspondence of a field extension, concluding with a brief discussion of the Krull topology. In Chapter 7, we discuss the Galois theory of equations. In Chapter 8, we view a field extension E of F as a vector space over F.
Chapter 9 and Chapter 10 are devoted to finite fields, although this material can be omitted in order to reach the topic of solvability by radicals more quickly.Mobius inversion is used in a few places, so an appendix has been included on this subject.
Part 3 of the book is entitled The Theory of Binomials. Chapter 11 covers the roots of unity and Wedderbum's theorem on finite division rings. We also briefly discuss the question of whether a given group is the Galois group of a field extension. In Chapter 12, we characterize cyclic extensions and splitting fields of binomials when the base field contains appropriate roots of unity.Chapter 13 is devoted to the question of solvability of a polynomial equation by radicals. (This chapter might make a convenient ending place in a graduate course.) In Chapter 14, we determine conditions that characterize the irreducibility of a binomial and describe the Galois group of a binomial. Chapter 15 briefly describes the theory of families of binomials--the so-called Kummer theory.
Sections marked with an asterisk may be skipped without loss of continuity. hanges for the Second Edition
Let me begin by thanking the readers of the first edition for their many helpful comments and suggestions.
For the second edition, I have gone over the entire book, and rewritten most of it, including the exercises. I believe the book has benefited significantly from a class testing at the beginning graduate level and at a more advanced graduate level.
I have also rearranged the chapters on separability and algebraic independence,feeling that the former is more important when time is of the essence. In my course, I generally touch only very lightly (or skip altogether) the chapter on algebraic independence, simply because of time constraints.
As mentioned earlier, as several readers have requested, 1 have added a chapter on Galois theory from an historical perspective.
A few additional topics are sprinkled throughout, such as a proof of the Fundamental Theorem of Algebra, a discussion of casus irreducibilis,Berlekamp's algorithm for factoring polynomials over Zp and natural and accessory irrationalities.
Thanks
I would like to thank my students Phong Le, Sunil Chetty, Timothy Choi and Josh Chan, who attended lectures on essentially the entire book and offeredmany helpful suggestions. I would also like to thank my editor, Mark Spencer,who puts up with my many requests and is most amiable.
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