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

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

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