
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.
Cloth
ISBN 0130115843

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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 14 are for an undergraduate course while
chapters 59 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).
NEW—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).
NEW—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 readerfriendly, 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, easytofollow 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. ___
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.
