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Full Description
Studying abstract algebra can be an adventure of awe-inspiring discovery. The subject need not be watered down nor should it be presented as if all students will become mathematics instructors. This is a beautiful, profound, and useful field which is part of the shared language of many areas both within and outside of mathematics.
To begin this journey of discovery, some experience with mathematical reasoning is beneficial. This text takes a fairly rigorous approach to its subject, and expects the reader to understand and create proofs as well as examples throughout.
The book follows a single arc, starting from humble beginnings with arithmetic and high-school algebra, gradually introducing abstract structures and concepts, and culminating with Niels Henrik Abel and Evariste Galois' achievement in understanding how we can—and cannot—represent the roots of polynomials.
The mathematically experienced reader may recognize a bias toward commutative algebra and fondness for number theory.
The presentation includes the following features:
Exercises are designed to support and extend the material in the chapter, as well as prepare for the succeeding chapters.
The text can be used for a one, two, or three-term course.
Each new topic is motivated with a question.
A collection of projects appears in Chapter 23.
Abstract algebra is indeed a deep subject; it can transform not only the way one thinks about mathematics, but the way that one thinks—period. This book is offered as a manual to a new way of thinking. The author's aim is to instill the desire to understand the material, to encourage more discovery, and to develop an appreciation of the subject for its own sake.
Contents
Preface
Symbols
1.Review of Sets, Functions, and Proofs
2.Introduction: A Number Game
3.Groups
4.Subgroups
5.Symmetry
6.Free Groups
7.Group Homomorphisms
8.Lagrange's Theorem
9.Special Types of Homomorphisms
10.Making Groups
11.Rings
12.Results on Commutative Rings
13.Vector Spaces
14.Polynomial Rings
15.Field Theory
16.Galois Theory
17.Direct Sums and Direct Products
18.The Structure of Finite Abelian Groups
19.Group Actions
20.Learning from Z
21.The Problems of the Ancients
22.Solvability of Polynomial Equations by Radicals
23.Projects
Bibliography
Index
