An Introduction to Groups, Rings, and Fields

Bruce Cooperstein




An Introduction to Groups, Rings, and Fields

An Introduction to Groups, Rings, and Fields
Bruce Cooperstein
University of California, Santa Cruz

ISBN-10: 0-9842071-4-7
ISBN-13: 978-0-9842071-4-5
440 Pages
©2012 Worldwide Center of Mathematics

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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.




Chapter 1: Preliminaries
1.1 Sets
1.2 Relations
1.3 Functions
1.4 Natural Numbers
1.5 Integers
1.6 Counting
1.7 Partially Ordered Sets

Chapter 2: Elementary Group Theory
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

Chapter 3: Rings
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

Chapter 4: Vector Spaces
4.1 Vector Spaces Over a Field F
4.2 Span and Independence
4.3 Bases and Dimension

Chapter 5: Composition Series and Solvable Groups
5.1 Compositions Series and the Jordan-Holder Theorem
5.2 Solvable Groups

Chapter 6: Fields
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




Bruce Cooperstein:
Bruce Cooperstein received his Ph.D. in mathematics from the University of Michigan in 1975. He has been on the faculty of the University of California, Santa Cruz continuously since 1975, obtaining the rank of full professor in 1989. He has won two prestigious awards, a W.K.Kellogg National Fellowship (1982-85) and a Pew National Scholarship for Carnegie Scholars (1999-2000). Bruce’s research areas include finite groups, groups of Lie Type, Lie geometries, incidence and Galois geometry. He is author of one of the first on-line course portfolios, Learning to Think Mathematically. and is the author of over 50 papers that have appeared in referred journals and proceedings of conferences. Bruce was a visiting Fellow of the Carnegie Foundation in Spring, 2007, and has also been involved in mathematics teacher professional development and mathematics education for over two decades.