A Concrete Approach to Abstract Algebra

A Concrete Approach to Abstract Algebra

From the Integers to the Insolvability of the Quintic

1st Edition - December 28, 2009

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  • Author: Jeffrey Bergen
  • eBook ISBN: 9780080958620

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A Concrete Approach to Abstract Algebra presents a solid and highly accessible introduction to abstract algebra by providing details on the building blocks of abstract algebra.It begins with a concrete and thorough examination of familiar objects such as integers, rational numbers, real numbers, complex numbers, complex conjugation, and polynomials. The author then builds upon these familiar objects and uses them to introduce and motivate advanced concepts in algebra in a manner that is easier to understand for most students. Exercises provide a balanced blend of difficulty levels, while the quantity allows the instructor a latitude of choices. The final four chapters present the more theoretical material needed for graduate study.This text will be of particular interest to teachers and future teachers as it links abstract algebra to many topics which arise in courses in algebra, geometry, trigonometry, precalculus, and calculus.

Key Features

  • Presents a more natural 'rings first' approach to effectively leading the student into the the abstract material of the course by the use of motivating concepts from previous math courses to guide the discussion of abstract algebra
  • Bridges the gap for students by showing how most of the concepts within an abstract algebra course are actually tools used to solve difficult, but well-known problems
  • Builds on relatively familiar material (Integers, polynomials) and moves onto more abstract topics, while providing a historical approach of introducing groups first as automorphisms
  • Exercises provide a balanced blend of difficulty levels, while the quantity allows the instructor a latitude of choices


Advanced undergraduate and graduate students studying mathematics, education, biology, chemistry, economics, environmental sciences, physics, computer science and engineering taking a course in abstract algebra

Table of Contents

  • 1. Introduction
    2. What This Book Is about and Who This Book Is for
    3. Proof and Intuition
    4. The Integers
    5. Rational Numbers and the Real Numbers
    6. The Complex Numbers
    7. The fundamental Theorem of Algebra
    8. The Integers Modulo n
    9. Group Theory
    10. Polynomials over the Integers and Rationals
    11. Roots of Polynomials of Degree Less than 5
    12. Rational Values of Trigonometric Functions
    13. Polynomials over Arbitrary Fields
    14. Difference Functions and Partial Fractions
    15. An Introduction to Linear Algebra and Vector Spaces
    16. Degrees and Galois Groups of Field Extensions
    17. Geometric Constructions
    18. The Insolvability of the Quintic
    19. Bibliography
    20. Index

Product details

  • No. of pages: 720
  • Language: English
  • Copyright: © Academic Press 2010
  • Published: December 28, 2009
  • Imprint: Academic Press
  • eBook ISBN: 9780080958620

About the Author

Jeffrey Bergen

Jeffrey Bergen (DePaul, Chicago), received his B.S. in Mathematics from Brooklyn College in 1976. He received his M.S. in 1977 and Ph.D. in 1981 from the University of Chicago. His DePaul career began in 1981, where he continues to do research in the branch of abstract algebra known as noncommutative ring theory. His research has received external support from the English Speaking Union, the National Science Foundation, and the National Security Agency. He has given lectures in 7 countries and co-authored papers with 16 mathematicians around the world. In 2001, he received the Excellence in Teaching Award from the College of Liberal Arts and Sciences and, in 2007, received their Cortelyou-Lowery Award for Excellence.

Affiliations and Expertise

DePaul University, Chicago, USA

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