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 Abstract AlgebraThird Edition
John A. Beachy and William D. Blair
Highly regarded by instructors in past editions
for its sequencing of topics as well as its concrete approach, slightly slower
beginning pace, and extensive set of exercises, the latest edition of Abstract
Algebra extends the thrust of the widely used earlier editions as it introduces
modern abstract concepts only after a careful study of important examples. Beachy and Blair’s clear narrative presentation responds to the needs of
inexperienced students who stumble over proof writing, who understand
definitions and theorems but cannot do the problems, and who want more examples
that tie into their previous experience. The authors introduce chapters by
indicating why the material is important and, at the same time, relating the new
material to things from the student’s background and linking the subject matter
of the chapter to the broader picture. Instructors will find the latest edition pitched at a suitable level of difficulty and will appreciate its gradual increase in the level of sophistication as the student progresses through the book. Rather than inserting superficial applications at the expense of important mathematical concepts, the Beachy and Blair solid, well-organized treatment motivates the subject with concrete problems from areas that students have previously encountered, namely, the integers and polynomials over the real numbers.
Supplementary material for instructors and
students available on the book’s Web site:
www.math.niu.edu/~beachy/abstract_algebra/ $65.95 list, 484 pages 10-digit ISBN: 1-57766-443-4 13-digit ISBN: 978-1-57766-443-7 © 2006 Instructor’s Solutions Manual available “The book has a great selection of exercises. There are enough good ones to make it possible to use the book several semesters in a row without repeating too much. They come in a nice mix from easy computations to warm the students up to more difficult theoretical problems. A few have solutions, but most don’t. I like this balance very much.” — Will Murray, California State University, Long Beach “I like the gradual introduction to abstraction by starting with examples rather than abstract groups or rings. Many nice examples, as well as good theorems often omitted from undergraduate courses.” — William M. McGovern, University of Washington “A well-written text with plenty of opportunity for students to get involved in the learning process.” — Francis T. Hannick, Minnesota State University, Mankato “This book moves from concreteness to abstraction more skillfully than any text I have ever seen. A completely convincing and student-oriented presentation of the ‘why’ of abstract algebra as well as the ‘how.’” — Vic Camillo, University of Iowa Table of Contents
1. Integers Divisors / Primes / Congruences / Integers Modulo n 2. Functions Functions / Equivalence Relations / Permutations 3. Groups Definition of a Group / Subgroups / Constructing Examples / Isomorphisms / Cyclic Groups / Permutation Groups / Homomorphisms / Cosets, Normal Subgroups, and Factor Groups 4. Polynomials Fields; Roots of Polynomials / Factors / Existence of Roots / Polynomials over Z, Q, R, and C 5. Commutative Rings Commutative Rings; Integral Domains / Ring Homomorphisms / Ideals and Factor Rings / Quotient Fields 6. Fields Algebraic Elements / Finite and Algebraic Extensions / Geometric Constructions / Splitting Fields / Finite Fields / Irreducible Polynomials over Finite Fields / Quadratic Reciprocity 7. Structure of Groups Isomorphism Theorems; Automorphisms / Conjugacy / Groups Acting on Sets / The Sylow Theorems / Finite Abelian Groups / Solvable Groups / Simple Groups 8. Galois Theory The Galois Group of a Polynomial / Multiplicity of Roots / The Fundamental Theorem of Galois Theory / Solvability by Radicals / Cyclotomic Polynomials / Computing Galois Groups 9. Unique Factorization Principal Ideal Domains / Unique Factorization Domains / Some Diophantine Equations Appendix Sets / Construction of the Number Systems / Basic Properties of the Integers / Induction / Complex Numbers / Solution of Cubic and Quartic Equations / Dimension of a Vector Space
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