世界图书出版公司出版书籍域论 _生活百科

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域论(世界图书出版公司出版书籍)【世界图书出版公司出版书籍域论】《域论》是由（美）罗曼编写，世界图书出版公司出版的一本书籍。
基本介绍书名：《域论》
作者：（美）罗曼
原版名称：《Field Theory》 Second Edition
ISBN：7-5100-3763-8, 978-7-5100-3763-4
页数：332页
定价：34RMB
出版社：世界图书出版公司
出版时间：2011年7月1日（第2版）
装帧：平装
开本：24开
正文语种：英语
尺寸：22.4 x 14.6 x 1.6 cm
重量：399g
ASIN：B0063CRT6W
主要内容《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prefacecontents0 preliminaries0.1 lattices0.2 groups0.3 the symmetric group0.4 rings0.5 integral domains0.6 unique factorization domains0.7 principal ideal domains0.8 euclidean domains0.9 tensor productsexercisespart i-field extensions1 polynomials1.1 polynomials over a ring1.2 primitive polynomials and irreducibility1.3 the division algorithm and its consequences1.4 splitting fields.1.5 the minimal polynomial1.6 multiple roots1.7 testing for irreducibilityexercises2 field extensions2.1 the lattice of subfields of a field2.2 types of field extensions2.3 finitely generated extensions2.4 simple extensions2.5 finite extensions2.6 algebraic extensions2.7 algebraic closures2.8 embeddings and their extensions.2.9 splitting fields and normal extensionsexercises3 embeddings and separability3.1 recap and a useful lemma3.2 the number of extensions: separable degree3.3 separable extensions3.4 perfect fields3.5 pure inseparability3.6 separable and purely inseparable closuresexercises4 algebraic independence4.1 dependence relations4.2 algebraic dependence4.3 transcendence bases4.4 simple transcendental extensionsexercisespart ii——-galois theory5 galois theory i: an historical perspective5.1 the quadratic equation5.2 the cubic and quartic equations5.3 higher-degree equations5.4 newton's contribution: symmetric polynomials5.5 vandermonde5.6 lagrange5.7 gauss5.8 back to lagrange5.9 galois5.10 a very brief look at the life of galois6 galois theory i1: the theory6.1 galois connections6.2 the galois correspondence6.3 who's closed?6.4 normal subgroups and normal extensions6.5 more on galois groups6.6 abelian and cyclic extensions*6.7 linear disjointnessexercises7 galois theory iii: the galois group of a polynomial7.1 the galois group of a polynomial7.2 symmetric polynomials7.3 the fundamental theorem of algebra.7.4 the discriminant of a polynomial7.5 the galois groups of some small-degree polynomialsexercises8 a field extension as a vector space8.1 the norm and the trace*8.2 characterizing bases*8.3 the normal basis theoremexercises9 finite fields i: basic properties9.1 finite fields redux9.2 finite fields as splitting fields9.3 the subfields of a finite field.9.4 the multiplicative structure of a finite field9.5 the galois group of a finite field9.6 irreducible polynomials over finite fields*9.7 normal bases*9.8 the algebraic closure of a finite fieldexercises10 finite fields i1: additional properties10.1 finite field arithmetic10.2 the number of irreducible polynomials10.3 polynomial functions10.4 linearized polynomialsexercises11 the roots of unity11.1 roots of unity11.2 cyclotomic extensions11.3 normal bases and roots of unity11.4 wedderburn's theorem11.5 realizing groups as galois groupsexercises12 cyclic extensions12.1 cyclic extensions12.2 extensions of degree char（f）exercises13 solvable extensions13.1 solvable groups13.2 solvable extensions13.3 radical extensions13.4 solvability by radicals13.5 solvable equivalent to solvable by radicals13.6 natural and accessory irrationalities13.7 polynomial equationsexercisespart iii——the theory of binomials14 binomials14.1 irreducibility14.2 the galois group of a binomial14.3 the independence of irrational numbersexercises15 families of binomials15.1 the splitting field15.2 dual groups and pairings15.3 kummer theoryexercisesappendix: mobius inversionpartially ordered setsthe incidence algebra of a partially ordered setclassical mobius inversionmultiplicative version of m6bius inversionreferencesindex前言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 EditionLet 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.ThanksI 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.

世界图书出版公司出版书籍 域论

世界图书出版公司出版书籍域论