Introduction to Abstract Algebra

From Rings, Numbers, Groups, and Fields to Polynomials and Galois Theory

Benjamin Fine, Anthony M. Gaglione, Gerhard Rosenberger

• Publisher: Johns Hopkins University Press
• Year: 2014
• eBook ISBN: 9781421411774
• Edition: 0
• Topic: Mathematics
• Subtopic: Abstract Algebra
• Index: Mathematics

A new approach to abstract algebra that eases student anxieties by building on fundamentals.

Introduction to Abstract Algebra presents a breakthrough approach to teaching one of math's most intimidating concepts. Avoiding the pitfalls common in the standard textbooks, Benjamin Fine, Anthony M. Gaglione, and Gerhard Rosenberger set a pace that allows beginner-level students to follow the progression from familiar topics such as rings, numbers, and groups to more difficult concepts.

Classroom tested and revised until students achieved consistent, positive results, this textbook is designed to keep students focused as they learn complex topics. Fine, Gaglione, and Rosenberger's clear explanations prevent students from getting lost as they move deeper and deeper into areas such as abelian groups, fields, and Galois theory.

This textbook will help bring about the day when abstract algebra no longer creates intense anxiety but instead challenges students to fully grasp the meaning and power of the approach.

Topics covered include:
• Rings
• Integral domains
• The fundamental theorem of arithmetic
• Fields
• Groups
• Lagrange's theorem
• Isomorphism theorems for groups
• Fundamental theorem of finite abelian groups
• The simplicity of A n for n5
• Sylow theorems
• The Jordan-Hölder theorem
• Ring isomorphism theorems
• Euclidean domains
• Principal ideal domains
• The fundamental theorem of algebra
• Vector spaces
• Algebras
• Field extensions: algebraic and transcendental
• The fundamental theorem of Galois theory
• The insolvability of the quintic

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• 1 Abstract Algebra and Algebraic Reasoning
• 2 Algebraic Preliminaries
• 3 Rings and the Integers
• 4 Number Theory and Unique Factorization
• 5 Fields: The Rationals, Reals and Complexes
• 6 Basic Group Theory
• 7 Factor Groups and the Group Isomorphism Theorems
• 8 Direct Products and Abelian Groups
• 9 Symmetric and Alternating Groups
• 10 Group Actions and Topics in Group Theory
• 11 Topics in Ring Theory
• 12 Polynomials and Polynomial Rings
• 13 Algebraic Linear Algebra
• 14 Fields and Field Extensions
• 15 A Survey of Galois Theory
• Bibliography
• Index