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An Introduction to Algebraic Topology
  • Language: en
  • Pages: 447

An Introduction to Algebraic Topology

A clear exposition, with exercises, of the basic ideas of algebraic topology. Suitable for a two-semester course at the beginning graduate level, it assumes a knowledge of point set topology and basic algebra. Although categories and functors are introduced early in the text, excessive generality is avoided, and the author explains the geometric or analytic origins of abstract concepts as they are introduced.

An Introduction to Homological Algebra
  • Language: en
  • Pages: 710

An Introduction to Homological Algebra

Graduate mathematics students will find this book an easy-to-follow, step-by-step guide to the subject. Rotman’s book gives a treatment of homological algebra which approaches the subject in terms of its origins in algebraic topology. In this new edition the book has been updated and revised throughout and new material on sheaves and cup products has been added. The author has also included material about homotopical algebra, alias K-theory. Learning homological algebra is a two-stage affair. First, one must learn the language of Ext and Tor. Second, one must be able to compute these things with spectral sequences. Here is a work that combines the two.

Advanced Modern Algebra
  • Language: en
  • Pages: 1012

Advanced Modern Algebra

  • Type: Book
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  • Published: 2002
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  • Publisher: Unknown

For two-term undergraduate level courses in Algebra. This text's organizing principle is the interplay between groups and rings, where rings includes the ideas of modules. It contains basic definitions, complete and clear theorems and gives attention to the topics of algebraic geometry, computers, homology and representations. More than merely a succession of definition theorem proofs, this text puts results and ideas in context so that students can appreciate why a certain topic is being studied and where definitions originate. *Coverage of topics not usually found in other texts - e.g. inverse and direct limits: Euclidean rings; Grobner bases; Ext and tor; Schreier-Neilsen theorem (subgroups of free groups are free); simplicity of PSL (2, q). *Numerous exercises. *Many examples and counter-examples. *Serious treatment of set theory - Reminds students what functions really are. *Early presentation of the basis theorem for finite abelian groups - Makes the proof of the basis theorem for finitely generated modules over PID's more digestible, allowing students to then see how that proof is translated into the language of modules. *Transition - To make the step from an undergraduat

Journey into Mathematics
  • Language: en
  • Pages: 386

Journey into Mathematics

Students learn how to read and write proofs by actually reading and writing them, asserts author Joseph J. Rotman, adding that merely reading about mathematics is no substitute for doing mathematics. In addition to teaching how to interpret and construct proofs, Professor Rotman's introductory text imparts other valuable mathematical tools and illustrates the intrinsic beauty and interest of mathematics. Journey into Mathematics offers a coherent story, with intriguing historical and etymological asides. The three-part treatment begins with the mechanics of writing proofs, including some very elementary mathematics--induction, binomial coefficients, and polygonal areas--that allow students to focus on the proofs without the distraction of absorbing unfamiliar ideas at the same time. Once they have acquired some geometric experience with the simpler classical notion of limit, they proceed to considerations of the area and circumference of circles. The text concludes with examinations of complex numbers and their application, via De Moivre's theorem, to real numbers.

Advanced Modern Algebra: Third Edition, Part 2
  • Language: en
  • Pages: 558

Advanced Modern Algebra: Third Edition, Part 2

This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study.

A First Course in Abstract Algebra
  • Language: en
  • Pages: 552

A First Course in Abstract Algebra

  • Type: Book
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  • Published: 2000
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  • Publisher: Unknown

For one-semester or two-semester undergraduate courses in Abstract Algebra. This new edition has been completely rewritten. The four chapters from the first edition are expanded, from 257 pages in first edition to 384 in the second. Two new chapters have been added: the first 3 chapters are a text for a one-semester course; the last 3 chapters are a text for a second semester. The new Chapter 5, Groups II, contains the fundamental theorem of finite abelian groups, the Sylow theorems, the Jordan-Holder theorem and solvable groups, and presentations of groups (including a careful construction of free groups). The new Chapter 6, Commutative Rings II, introduces prime and maximal ideals, unique factorization in polynomial rings in several variables, noetherian rings and the Hilbert basis theorem, affine varieties (including a proof of Hilbert's Nullstellensatz over the complex numbers and irreducible components), and Grobner bases, including the generalized division algorithm and Buchberger's algorithm.

Advanced Modern Algebra
  • Language: en
  • Pages: 570

Advanced Modern Algebra

This book is the second part of the new edition of Advanced Modern Algebra (the first part published as Graduate Studies in Mathematics, Volume 165). Compared to the previous edition, the material has been significantly reorganized and many sections have been rewritten. The book presents many topics mentioned in the first part in greater depth and in more detail. The five chapters of the book are devoted to group theory, representation theory, homological algebra, categories, and commutative algebra, respectively. The book can be used as a text for a second abstract algebra graduate course, as a source of additional material to a first abstract algebra graduate course, or for self-study.

Learning Modern Algebra
  • Language: en
  • Pages: 481

Learning Modern Algebra

  • Type: Book
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  • Published: 2013
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  • Publisher: MAA

Learning Modern Algebra aligns with the CBMS Mathematical Education of Teachers–II recommendations, in both content and practice. It emphasizes rings and fields over groups, and it makes explicit connections between the ideas of abstract algebra and the mathematics used by high school teachers. It provides opportunities for prospective and practicing teachers to experience mathematics for themselves, before the formalities are developed, and it is explicit about the mathematical habits of mind that lie beneath the definitions and theorems.This book is designed for prospective and practicing high school mathematics teachers, but it can serve as a text for standard abstract algebra courses a...

Galois Theory
  • Language: en
  • Pages: 115

Galois Theory

This text offers a clear, efficient exposition of Galois Theory with exercises and complete proofs. Topics include: Cardano's formulas; the Fundamental Theorem; Galois' Great Theorem (solvability for radicals of a polynomial is equivalent to solvability of its Galois Group); and computation of Galois group of cubics and quartics. There are appendices on group theory and on ruler-compass constructions. Developed on the basis of a second-semester graduate algebra course, following a course on group theory, this book will provide a concise introduction to Galois Theory suitable for graduate students, either as a text for a course or for study outside the classroom.

An Introduction to Homological Algebra
  • Language: en

An Introduction to Homological Algebra

The landscape of homological algebra has evolved over the last half-century into a fundamental tool for the working mathematician. This book provides a unified account of homological algebra as it exists today. The historical connection with topology, regular local rings, and semi-simple Lie algebras are also described. This book is suitable for second or third year graduate students. The first half of the book takes as its subject the canonical topics in homological algebra: derived functors, Tor and Ext, projective dimensions and spectral sequences. Homology of group and Lie algebras illustrate these topics. Intermingled are less canonical topics, such as the derived inverse limit functor lim1, local cohomology, Galois cohomology, and affine Lie algebras. The last part of the book covers less traditional topics that are a vital part of the modern homological toolkit: simplicial methods, Hochschild and cyclic homology, derived categories and total derived functors. By making these tools more accessible, the book helps to break down the technological barrier between experts and casual users of homological algebra.