An Introduction to Groups, Rings, and Fields

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Bruce Cooperstein– University of California, Santa Cruz

ISBN-10: 0-9885572-0-7
ISBN-13: 978-0-9842071-4-5
440 Pages
©2019 Worldwide Center of Mathematics, LLC

Digital | $14.95

 

contents

Introduction

This is a digital textbook for a first course (sequence of courses) in Abstract Algebra covering the essentials of groups, rings and fields. The book is not an electronic version of a traditional print textbook but rather makes use of the digital environment to enhance student learning. One way this is achieved is by spiraling through the material, periodically returning to previous concepts to reinforce students' understanding. Specifically, every section begins with a subsection entitled: What do I Need to Know, where all the concepts that will be used are listed with links to their definitions. Also, throughout the text, all fundamental concepts are linked to their definitions so that students can return over and over again as needed to review them. Also, all citations to lemmas, propositions, theorems and corollaries throughout the text are linked to their statements and proofs.

Contents

  • 1.1 Sets
  • 1.2 Relations
  • 1.3 Functions
  • 1.4 Natural Numbers
  • 1.5 Integers
  • 1.6 Counting
  • 1.7 Partially Ordered Sets
  • 2.1 Definition of a Group and Examples
  • 2.2 Basic Properties of Groups
  • 2.3 Subgroups
  • 2.4 Cyclic Groups
  • 2.5 The Symmetric Group
  • 2.6 Products of Subgroups
  • 2.7 Normal Subgroups
  • 2.8 Quotient Groups
  • 2.9 Homomorphisms
  • 2.10 Automorphisms of Groups
  • 2.11 Alternating Groups
  • 2.12 Group Actions
  • 2.13 The Class Equation
  • 2.14 Sylow’s Theorems
  • 2.15 Direct Products of Groups
  • 2.16 Finite Abelian Groups
  • 3.1 Introduction to Rings
  • 3.2 Integral Domains
  • 3.3 Polynomial Rings
  • 3.4 Homomorphisms and Ideals
  • 3.5 Quotient Rings
  • 3.6 Prime and Maximal Ideals in Commutative Rings
  • 3.7 Field of Fractions of an Integral Domain
  • 3.8 Principal Ideal Domains and Euclidean Domains
  • 3.9 Polynomials Over Fields
  • 3.10 The Gaussian Integers
  • 3.11 Unique Factorization Domains
  • 4.1 Vector Spaces Over a Field F
  • 4.2 Span and Independence
  • 4.3 Bases and Dimension
  • 5.1 Compositions Series and the Jordan-Holder Theorem
  • 5.2 Solvable Groups
  • 6.1 Extensions of Fields
  • 6.2 Splitting Fields and Roots of Polynomials
  • 6.3 Finite Fields
  • 6.4 Constructible Numbers
  • 6.5 Galois Theory
  • 6.6 Cyclotomic Polynomials and Extensions
  • 6.7 Solvability by Radicals
  • 6.8 Cubic and Quartic Polynomials