[Book Cover]

A First Course in Abstract Algebra, 2/e

Joseph Rotman, University of Illinois

Coming February, 2000 by Prentice Hall Engineering/Science/Mathematics

Copyright 2000, 488 pp.
ISBN 0-13-011584-3

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For undergraduate and graduate courses in Abstract Algebra. This new edition is more a new book than a revision. The four chapters from the first edition are heavily revised and expanded. Five new chapters (running nearly 100 pages each) on graduate topics have been added. Text now has two years of algebra from which to choose.


NEW—Fits both undergraduate and graduate courses—Chapters 1-4 are for an undergraduate course while chapters 5-9 are for a full year of graduate algebra.
NEW—Transition chapter to more advanced material—Ties preceding chapters together to solve famous classical problems and provides a transition from undergraduate to graduate material (Ch. 4).

  • Allows students to see how concepts can be applied to fascinating problems. Eases students into advanced material.
NEW—Modules—Includes coverage of categories and functors, free modules, projectives and injectives, as well as applications of Zorn's Lemma (Ch.5).
NEW—Algebras—Presents noncommutative rings, semisimple rings, tensor products, tensor algebras, and exterior algebras (Ch. 6).
Principal Ideal Domains—Introduces presentations of modules, rational canonical forms, Jordan canonical forms, as well as the Smith normal form for modules over euclidean rings, and infinite abelian groups (Ch. 7).
NEW—Group theory—Introduces Sylow theorem, simple group, free groups and presentations, semidirect products, and the extension problem and its connection with low dimensional group cohomology, among other topics (Ch. 8).
Polynomials in several variables—Presents Hilbert basis theorem, nullstellensatz, generalized division algorithm and Grobner bases, varieties, localization and Spec, among others (Ch. 9).
Number theory—Presents concepts such as induction, factorization into primes, binomial coefficients and DeMoivre's Theorem (Ch. 1).
  • Allows students to learn to write proofs in a familiar context without the added complication of simultaneously absorbing abstract new ideas.
Permutations presented before groups.
  • Allows students to master more basic concepts before moving on to more difficult material. Ex. ___
Groups treated before rings.
  • Gives students a reader-friendly, efficient presentation of material. Ex. ___
History and etymology—Interwoven throughout the text.
  • Provides students with an interesting and relevant history of ideas rather than gossip and personal details.
Complete and clear proofs—Presented throughout the text.
  • Give students clear, easy-to-follow models. Ex. ___
Exercises—Appear throughout the book and are solved in a separate solution manual (written by Professor Rotman) for instructors.
  • Provide students with ample opportunity to apply the concepts they have learned. Allow instructors to quickly check student work.
Examples—Include interesting applications, e.g., writing numbers in base 3 solves certain weighing problems; the Chinese Remainder Theorem is used in computations with an ancient Mayan calendar.
  • Allow students to see concepts applied in unorthodox and surprising ways, generating interest and encouraging critical thinking. Ex. ___

Table of Contents
    1. Number Theory.

      Introduction. Binomial Coefficients. Greatest Common Divisors. The Fundamental Theorem of Arithmetic. Congruences. Dates and Days.

    2. Groups.

      Functions. Permutations. Groups. Lagrage's Theorem. Homomorphisms. Quotient Groups. Group Actions. Counting with Groups.

    3. Commutative Rings.

      First Properties. Fields. Polynomials. Greatest Common Divisors. Unique Factorization. Homomorphisms. Irreducibility. Quotient Rings and Finite Fields. Officers, Fertilizer, and a Line at Infinity.

    4. Goodies.

      Linear Algebra. Euclidean Constructions. Classical Formulas. Insolvability of the General Quintic. Epilogue.

    5. Modules.

      Application's of Zorn's Lemma. Modules. Categories and Functors. Free Modules, Projectives, and Injectives.

    6. Algebras.

      Noncommutative Rings. Semisimple Rings. Tensor Products. Algebras.

    7. Principal Ideal Domains.
    8. Group Theory II.
    9. Polynomials In Several Variables.


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