CONTENTS
*
Preface
Chapter 1 Why Abstract Algebra?
History of Algebra. New Algebras. Algebraic Structures. Axioms and Axiomatic Algebra.
Abstraction in Algebra.
Chapter 2 Operations
Operations on a Set. Properties of Operations.
Chapter 3 The Definition of Groups
Groups. Examples of Infinite and Finite Groups. Examples of Abelian and Nonabelian
Groups. Group Tables.
Theory of Coding: Maximum-Likelihood Decoding.
Chapter 4 Elementary Properties of Groups
Uniqueness of Identity and Inverses. Properties of Inverses.
Direct Product of Groups.
Chapter 5 Subgroups
Definition of Subgroup. Generators and Defining Relations.
Cayley Diagrams. Center of a Group. Group Codes; Hamming Code
.
Chapter 6 Functions
Injective, Surjective, Bijective Function. Composite and Inverse of Functions.
Finite-State Machines. Automata and Their Semigroups.
Chapter 7 Groups of Permutations
Symmetric Groups. Dihedral Groups.
An Application of Groups to Anthropology.
Chapter 8 Permutations of a Finite Set
Decomposition of Permutations into Cycles. Transpositions. Even and Odd Permutations.
Alternating Groups.
Chapter 9 Isomorphism
The Concept of Isomorphism in Mathematics. Isomorphic and Nonisomorphic Groups.
Cayley’s Theorem.
Group Automorphisms
.
Chapter 10 Order of Group Elements
Powers/Multiples of Group Elements. Laws of Exponents. Properties of the Order of Group Elements.
Chapter 11 Cyclic Groups
Finite and Infinite Cyclic Groups. Isomorphism of Cyclic Groups. Subgroups of Cyclic
Groups.
Chapter 12 Partitions and Equivalence Relations
Chapter 13 Counting Cosets
Lagrange’s Theorem and Elementary Consequences.
Survey of Groups of Order ≤ 10.
Number of Conjugate Elements. Group Acting on a Set.
Chapter 14 Homomorphisms
Elementary Properties of Homomorphisms. Normal Subgroups. Kernel and Range.
Inner Direct Products. Conjugate Subgroups.
Chapter 15 Quotient Groups
Quotient Group Construction. Examples and Applications.
The Class Equation. Induction on the Order of a Group.
Chapter 16 The Fundamental Homomorphism Theorem
Fundamental Homomorphism Theorem and Some Consequences.
The Isomorphism Theorems. The Correspondence Theorem. Cauchy’s Theorem. Sylow
Subgroups. Sylow’s Theorem. Decomposition Theorem for Finite Abelian Groups
.
Chapter 17 Rings: Definitions and Elementary Properties
Commutative Rings. Unity. Invertibles and Zero-Divisors. Integral Domain. Field.
Chapter 18 Ideals and Homomorphisms
Chapter 19 Quotient Rings
Construction of Quotient Rings. Examples. Fundamental Homomorphism Theorem and
Some Consequences. Properties of Prime and Maximal Ideals.
Chapter 20 Integral Domains
Characteristic of an Integral Domain. Properties of the Characteristic. Finite Fields.
Construction of the Field of Quotients.
Chapter21 The Integers
Ordered Integral Domains. Well-ordering. Characterization of
Up to Isomorphism.
Mathematical Induction. Division Algorithm.
Chapter 22 Factoring into Primes
Ideals of Z. Properties of the GCD. Relatively Prime Integers. Primes. Euclid’s Lemma.
Unique Factorization.
Chapter 23 Elements of Number Theory (Optional)
Properties of Congruence. Theorems of Fermât and Euler. Solutions of Linear Congruences.
Chinese Remainder Theorem.
Wilson’s Theorem and Consequences. Quadratic Residues. The Legendre Symbol.
Primitive Roots.
Chapter 24 Rings of Polynomials
Motivation and Definitions. Domain of Polynomials over a Field. Division Algorithm.
Polynomials in Several Variables. Fields of Polynomial Quotients.
Chapter 25 Factoring Polynomials
Ideals of F[x]. Properties of the GCD. Irreducible Polynomials. Unique factorization.
Euclidean Algorithm.
Chapter 26 Substitution in Polynomials
Roots and Factors. Polynomial Functions. Polynomials over Q
Eisenstein’s Irreducibility Criterion.
Polynomials over the Reals. Polynomial Interpolation.
Chapter 27 Extensions of Fields
Algebraic and Transcendental Elements. The Minimum Polynomial. Basic Theorem on
Field Extensions.
Chapter 28 Vector Spaces
Elementary Properties of Vector Spaces. Linear Independence. Basis. Dimension. Linear Transformations.
Chapter29 Degrees of Field Extensions
Simple and Iterated Extensions. Degree of an Iterated Extension.
Fields of Algebraic Elements. Algebraic Numbers. Algebraic Closure.
Chapter 30 Ruler and Compass
Constructible Points and Numbers. Impossible Constructions.
Constructible Angles and Polygons.
Chapter 31 Galois Theory: Preamble
Multiple Roots. Root Field. Extension of a Field. Isomorphism.
Roots of Unity. Separable Polynomials. Normal Extensions.
Chapter 32 Galois Theory: The Heart of the Matter
Field Automorphisms. The Galois Group. The Galois Correspondence. Fundamental
Theorem of Galois Theory.
Computing Galois Groups.
Chapter 33
Solving Equations by Radicals
Radical Extensions. Abelian Extensions. Solvable Groups. Insolvability of the Quin tic.
Appendix A Review of Set Theory
Appendix B Review of the Integers
Appendix C Review of Mathematical Induction
Answers to Selected Exercises
Index
· · · · · · (
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0 有用 舒尔霍夫 2024-05-01 01:21:46 英国
很适合初学诶,小时候怎么没看,现在看已经没用了哈哈哈。。把一些基本的东西讲得其实很深入
0 有用 北府少将 2016-07-04 10:30:41
还有三个chapter今天实在不想看了,换本书看看再来读剩下的,换本书。好想去图书馆借数学书来看木哈哈
1 有用 〇 2018-06-14 17:13:34
非常适合自学。最后利用域的扩张来建模尺规作图和方程是否根式可解,感受到代数结构把不同领域的世纪难题联系起来,并精妙求解,可以说是一种超高级享受了。
1 有用 zhuyan 2012-04-27 14:07:11
A good elementary introduction to abstract algebra
6 有用 Backwater 2015-06-12 12:29:07
亲测十分简单 读者友好 尽管有些apporaches确实挺猥琐的 十分适合复习补充使用或者当主要教材用 传说中庙堂之上的数学系人才看不起的教材...
0 有用 舒尔霍夫 2024-05-01 01:21:46 英国
很适合初学诶,小时候怎么没看,现在看已经没用了哈哈哈。。把一些基本的东西讲得其实很深入
0 有用 未见青山老 2024-03-30 23:21:59 山东
习题差点意思...
0 有用 SimpleLine 2023-11-10 11:55:35 上海
写得很清楚
0 有用 yar2001 2022-12-27 23:57:51 广东
这本书挺适合自学,介绍了抽象代数的来源和作用,以及和其它数学学科的联系。我之前接触到的数学,主要是对观察到的现象提取共性、屏蔽细节后,所泛化出一套抽象工具,让规律可以在不同事物中被充分认识、演绎和利用,拓宽了人类观测和处理到的信息维度。抽象代数也比较类似,但它是对数学结构本身的抽象,在提炼了各类代数系统中的共性后,在抽象上形成了又一层抽象。 读到第16章,熟悉了 Group、Subgroup、Fu... 这本书挺适合自学,介绍了抽象代数的来源和作用,以及和其它数学学科的联系。我之前接触到的数学,主要是对观察到的现象提取共性、屏蔽细节后,所泛化出一套抽象工具,让规律可以在不同事物中被充分认识、演绎和利用,拓宽了人类观测和处理到的信息维度。抽象代数也比较类似,但它是对数学结构本身的抽象,在提炼了各类代数系统中的共性后,在抽象上形成了又一层抽象。 读到第16章,熟悉了 Group、Subgroup、Function、Isomorphism、Homomorphism 等概念及其常用符号,对之后阅读 Category Theory 有关的教材文本有所帮助。不过,抽代的实际应用牵扯到相关的领域知识,我不搞这方面的研究,对体悟“数学的美感”和能不能解五次方程也不太感兴趣,实用性有限,先浅尝辄止吧。 (展开)
2 有用 狄拉克之旋 2022-04-05 21:56:36
伽罗瓦理论部分处理得不太干净利落,算是一点小小的缺陷吧。 习题设计的非常好(但是部分习题有错误),希望国内能赶快引进吧,dover出的这个小开本翻起来是真的不方便。