A Concrete Approach to Abstract Algebra: From the Integers to the Insolvability of the Quintic

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Academic Press, Dec 28, 2009 - Mathematics - 720 pages
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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.

  • 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
 

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Contents

Chapter 1 What This Book Is about and Who This Book Is for
1
Chapter 2 Proof and Intuition
19
Chapter 3 The Integers
61
Chapter 4 The Rational Numbers and the Real Numbers
97
Chapter 5 The Complex Numbers
137
Chapter 6 The Fundamental Theorem of Algebra
189
Chapter 7 The Integers Modulo n
227
Chapter 8 Group Theory
265
Chapter 11 Rational Values of Trigonometric Functions
423
Chapter 12 Polynomials over Arbitrary Fields
437
Chapter 13 Difference Functions and Partial Fractions
487
Chapter 14 An Introduction to Linear Algebra and Vector Spaces
527
Chapter 15 Degrees and Galois Groups of Field Extensions
573
Chapter 16 Geometric Constructions
623
Chapter 17 Insolvability of the Quintic
645
Bibliography
685

Chapter 9 Polynomials over the Integers and Rationals
365
Chapter 10 Roots of Polynomials of Degree Less than 5
411

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About the author (2009)

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.

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