抽象代數(shù)

前言

Algebra is used by virtually all mathematicians, be they analysts, combinatorists, computerscientists, geometers, logicians, number theorists, or topologists. Nowadays, everyoneagrees that some knowledge of linear algebra, groups, and commutative rings is necessary,and these topics are introduced in undergraduate courses. We continue their study.　 This book can be used as a text for the first year of graduate algebra, but it is much morethan that. It can also serve more advanced graduate students wishing to learn topics ontheir own; while not reaching the frontiers, the book does provide a sense of the successesand methods arising in an area. Finally, this is a reference containing many of the standardtheorems and definitions that users of algebra need to know. Thus, the book is not only anappetizer, but a hearty meal as well.　 Let me now address readers and instructors who use the book as a text for a beginninggraduate course. If I could assume that everyone had already read my book, A First Coursein Abstract Algebra, then the prerequisites for this book would be plain. But this is not arealistic assumption; different undergraduate courses introducing abstract algebra abound,as do texts for these comes. For many, linear algebra concentrates on matrices and vectorspaces over the real numbers, with an emphasis on computing solutions of linear systemsof equations; other courses may treat vector spaces over arbitrary fields, as well as Jordanand rational canonical forms. Some courses discuss the Sylow theorems; some do not;some comes classify finite fields; some do not.　 To accommodate readers having different backgrounds, the first three chapters containmany familiar results, with many proofs merely sketched. The first chapter contains thefundamental theorem of arithmetic, congruences, De Moivre's theorem, roots of unity,cyclotomic polynomials, and some standard notions of set theory, such as equivalencerelations and verification of the group axioms for symmetric groups. The next two chap-ters contain both familiar and unfamiliar material. "New" results, that is, results rarelytaught in a first course, have complete proofs, while proofs of "old" results are usuallysketched. In more detail, Chapter 2 is an introduction to group theory, reviewing permuta-tions, Lagrange's theorem, quotient groups, the isomorphism theorems, and groups actingon sets.

內(nèi)容概要

這批教材普遍具有以下特點(diǎn)：(1)基本上是近3年出版的，在國際上被廣泛使用，在同類教材中具有相當(dāng)?shù)臋?quán)威性；(2)高版次，歷經(jīng)多年教學(xué)實(shí)踐檢驗(yàn)，內(nèi)容翔實(shí)準(zhǔn)確、反映時(shí)代要求；(3)各種教學(xué)資源配套整齊，為師生提供了極大的便利；(4)插圖精美、豐富，圖文并茂，與正文相輔相成；(5)語言簡練、流暢、可讀性強(qiáng)，比較適合非英語國家的學(xué)生閱讀。

作者簡介

作者：(美國)羅特曼(Rotman,J.J.)

書籍目錄

Preface EtymologySpecial NotationChapter I Things Past1.1. Some Number Theory1.2. Roots of Unity1.3. Some Set Theory Chapter 2 Groups I2.1. Introduction2.2. Permutations2.3. Groups2.4. Lagrange's Theorem2.5. Homomorphisms2.6. Quotient Groups2.7. Group ActionsChapter 3 Commutative Rings I3.1. Introduction3.2. First Properties3.3. Polynomials3.4. Greatest Common Divisors3.5. Homomorphisms3.6. Euclidean Rings3.7. Linear AlgebraVector SpacesLinear Transformations3.8. Quotient Rings and Finite FieldsChapter 4 Fields4.1. Insolvability of the QuinticFormulas and Solvability by RadicalsTranslation into Group Theory4.2. Fundamental Theorem of Galois TheoryChapter 5 Groups II5.1. Finite Abelian GroupsDirect SumsBasis TheoremFundamental Theorem5.2. The Sylow Theorems5.3. The Jordan-Hilder Theorem5.4. Projective Unimodular Groups5.5. Presentations5.6. The Neilsen-Schreier TheoremChapter 6 Commutative Rings H6.1. Prime Ideals and Maximal Ideals6.2. Unique Factorization Domains6.3. Noetherian Rings6.4. Applications of Zom's Lemma6.5. Varieties6.6. Gr6bner BasesGeneralized Division AlgorithmBuchberger's AlgorithmChapter 7 Modules and Categories7.1. Modules7.2. Categories7.3. Functors7.4. Free Modules, Projectives, and Injectives7.5. Grothendieck Groups7.6. LimitsChapter 8 Algebras8.1. Noncommutative Rings8.2. Chain Conditions8.3. Semisimple Rings8.4. Tensor Products8.5. Characters8.6. Theorems of Burnside and of FrobeniusContentsChapter 9 Advanced Linear Algebra9.1. Modules over PIDs9.2. Rational Canonical Forms9.3. Jordan Canonical Forms9.4. Smith Normal Forms9.5. Bilinear Forms9.6. Graded Algebras9.7. Division Algebras9.8. Exterior Algebra9.9. Determinants9.10. Lie AlgebrasChapter 10 Homology10.1. Introduction10.2. Semidirect Products10.3. General Extensions and Cohomology10.4. Homology Functors10.5. Derived Functors10.6. Ext and Tor10.7. Cohomology of Groups10.8. Crossed Products10.9. Introduction to Spectral SequencesChapter 11 Commutative Rings III11.1. Local and Global11.2. Dedekind RingsIntegralityNullstellensatz ReduxAlgebraic IntegersCharacterizations of Dedekind RingsFinitely Generated Modules over Dedekind Rings11.3. Global Dimension11.4. Regular Local RingsAppendixThe Axiom of Choice and Zorn's LemmaBibliographyIndex

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