抽象代數(shù)基本教程

內(nèi)容概要

　　本書是一部介紹抽象代數(shù)的入門書籍。假定學(xué)生了解了微積分和線性代數(shù)，并且理論大都在書中以例子和練習(xí)的形式出現(xiàn)。該書旨在教給學(xué)生盡可能多的群，環(huán)，以及域理論，最大特點(diǎn)是包含了較多的扎實(shí)的基礎(chǔ)部分，這些部分對(duì)于更進(jìn)一步的學(xué)習(xí)代數(shù)是有很大的幫助的。為了滿足更多讀者的要求，本書包含了很多有關(guān)拓?fù)渲械耐{(diào)群，同調(diào)群的計(jì)算以加深對(duì)因子群的理解。書中內(nèi)容淺顯易懂，為了將同調(diào)群講述的更加清楚，在第6版的基礎(chǔ)上減少了自動(dòng)機(jī)，二進(jìn)制線性密碼以及部分代數(shù)結(jié)構(gòu)。書后面附有不少練習(xí)，這些加深學(xué)生對(duì)內(nèi)容的理解。　　目次：群與子群；排列，陪集和直積；同態(tài)和因子群；環(huán)和域；理想和因子環(huán)；擴(kuò)展域；高等群論；拓?fù)淙?；因子分解；自同態(tài)和伽羅瓦理論。

作者簡(jiǎn)介

作者：(美國(guó))菲來(lái) (Fraleigh.J.B)

書籍目錄

Instructor's PrefaceStudent's Preface Dependence Chart Sets and RelationsⅠ GROUPS AND SUBGROUPSIntroduction and ExamplesBinary OperationsIsomorphic Binary StructuresGroupsSubgroupsCyclic GroupsGenerating Sets and Cayley DigraphsⅡ PERMUTATIONS, COSETS, AND DIRECT PRODUCTSGroups of PermutationsOrbits, Cycles, and the Alternating GroupsCosets and the Theorem of LagrangeDirect Products and Finitely Generated Abelian GroupsPlane IsometriesⅢ HOMOMORPHISMS AND FACTOR GROUPSHomomorphismsFactor GroupsFactor-Group Computations and Simple GroupsGroup Action on a SetApplications of G-Sets to CountingⅣ RINGS AND FIELDSRings and FieldsIntegral DomainsFermat's and Euler's TheoremsThe Field of Quotients of an Integral Domain Rings of PolynomialsFactorization of Polynomials over a FieldNoncommutative ExamplesOrdered Rings and FieldsⅤ IDEALS AND FACTOR RINGSHomomorphisms and Factor RingsPrime and Maximal IdealsGrobner Bases for IdealsⅥ EXTENSION FIELDSIntroduction to Extension FieldsVector SpacesAlgebraic ExtensionsGeometric Constructions Finite FieldsⅦ ADVANCED GROUP THEORYIsomorphism TheoremsSeries of GroupsSylow TheoremsApplications of the Sylow TheoryFree Abelian GroupsFree GroupsGroup PresentationsⅧ GROUPS IN TOPOLOGYSimplicial Complexes and Homology GroupsComputations of Homology GroupsMore Homology Computations and ApplicationsHomological AlgebraⅨ FACTORIZATIONUnique Factorization DomainsEuclidean DomainsGaussian Integers and Multiplicative NormsⅩ AUTOMORPHISMS AND GALOIS THEORYAutomorphisms of FieldsThe Isomorphism Extension TheoremSplitting FieldsSeparable ExtensionsTotally Inseparable Extensions Galois TheoryIllustrations of Galois TheoryCyclotomic ExtensionsInsolvability of the QuinticAppendix: Matrix AlgebraBibliographyNotationsAnswers to Odd-Numbered Exercises Not Asking for Definitions or ProofsIndex

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