包邮抽象代数基础教程(英文版 ·原书第8版)

目　录教师前言学生前言第0章　集合和关系　1第1章　群和子群　121　二元运算　122　群　223　交换群的例子　374　非交换群的例子　485　子群　646　循环群　747　生成集和凯莱有向图　85第2章　群结构　938　置换群　93目　录教师前言学生前言第0章　集合和关系　1第1章　群和子群　121　二元运算　122　群　223　交换群的例子　374　非交换群的例子　485　子群　646　循环群　747　生成集和凯莱有向图　85第2章　群结构　938　置换群　939　有限生成交换群　10610　陪集和拉格朗日定理　11711　平面等距变换　126第3章　同态和商群　13512　商群　13513　商群计算和单群　14414　群在集合上的作用　15715　G集在计数中的应用　168第4章　群论进阶　17316　同构定理　17317　西罗定理　17818　群列　18719　自由交换群　19820　自由群　20621　群的表现　212第5章　环和域　22122　环和域的概念　22123　整环　23124　费马定理和欧拉定理　23925　加密　245第6章　环和域的构造　25126　整环的商域　25127　多项式环　25928　域上多项式的因式分解　27129　代数编码理论　28330　同态和商环　29131　素理想和极大理想　29932　非交换例子　308第7章　交换代数　31833　向量空间　31834　唯一分解整环　32835　欧几里得整环　34136　数论　34837　代数几何　35638　理想的Gr?bner基　363第8章　扩域　37239　扩域介绍　37240　代数扩张　38241　几何构造　39342　有限域　401第9章　伽罗瓦理论　40743　伽罗瓦理论导引　40744　分裂域　41745　可分扩张　42746　伽罗瓦理论主要定理　43647　伽罗瓦理论的描述　44548　分圆扩张　45349　五次方程的不可解性　459附录：矩阵代数　467参考文献　472记号　475部分习题答案　475Contents教师前言学生前言 0 SetsandRelations 1 I GROUPS AND SUBGROUPS 11 1 BinaryOperations 11 2 Groups 19 3 AbelianExamples 32 4 NonabelianExamples 39 5 Subgroups 52 6 CyclicGroups 61 7 GeneratingSetsandCayleyDigraphs 70 II STRUCTURE OF GROUPS 77 8 GroupsofPermutations 77 9 FinitelyGeneratedAbelianGroups 88 10 CosetsandtheTheoremofLagrange 97 11 .PlaneIsometries 105 III HOMOMORPHISMSAND FACTOR GROUPS 11312 FactorGroups 113 13 Factor-GroupComputationsand SimpleGroups 121 iii Contents .14 Group Action on a Set 132 .15 Applications of G-SetstoCounting 140 IV ADVANCED GROUP THEORY 145 16 Isomorphism Theorems 145 17 Sylow Theorems 149 18 Series ofGroups 157 19 Free Abelian Groups 166 20 Free Groups 172 21 Group Presentations 177 V RINGS AND FIELDS 185 22 Rings and Fields 185 23 Integral Domains 194 24 Fermat’s and Euler’sTheorems 200 25 Encryption 205 VI CONSTRUCTING RINGS AND FIELDS 211 26 TheFieldof Quotientsof anIntegral Domain 211 27 Rings of Polynomials 218 28 Factorization ofPolynomials over a Field 228 29 .AlgebraicCoding Theory 237 30 Homomorphisms andFactor Rings 243 31 Prime and MaximalIdeals 250 32 .Noncommutative Examples 258 VII COMMUTATIVE ALGEBRA 267 33 Vector Spaces 267 34 UniqueFactorization Domains 275 35 Euclidean Domains 286 36 Number Theory 292 37 .Algebraic Geometry 297 38 .Gr¨obner Basesfor Ideals 303 VIII EXTENSION FIELDS 311 39 IntroductiontoExtensionFields 311 40 AlgebraicExtensions 319 41 .GeometricConstructions 328 42 Finite Fields 335 Contents v IX GALOIS THEORY 341 43 Introductionto GaloisTheory 341 44 SplittingFields 349 45 SeparableExtensions 357 46 Galois Theory 364 47 Illustrations of Galois Theory 372 48 Cyclotomic Extensions 378 49 Insolvabilityof theQuintic 384 Appendix: Matrix Algebra 391 Bibliography 395 Notations 397 Answersto Odd-NumberedExercises Not Asking for De.nitions or Proofs 401 . Notrequiredfortheremainderofthetext. . This sectionisa prerequisite forSections17 and36only.